While I don't think this "sanctions" abortions per se, I think it is good to draw attention to this passage (and others like Exod 21:22-25) insofar as they show that all the "life begins at conception" etc. rhetoric of the religious right is not as clean-cut as they would have us believe.

That said, the context is that of divination---it is the deity, YHWH, who judges and makes the elixir effectual, not the elixir itself. In other words, this isn't about voluntary (or even forced) abortion, but ritual divination. It's more akin to tossing a woman into a pond to see if she's a witch, or casting lots to determine who is telling a lie.

If a wife is suspected by her husband of having an extramarital affair but there were no witnesses to the act (Num. 5:13), what was the priest to do?

If there were witnesses (Deut. 17:6), the women and the “other man” would be put to death (Lev. 20:10).

Since there were no witnesses, the woman was not condemned as an adulterer. Jesus himself presided over just such a case (John 8:10-11).

If the husband was still jealous he could bring his wife and a tenth of an ephah of finely ground barley (Lev. 2:1-3, Lev. 6:14-20) to the priest (Num. 5:15).

Rabbi’s have considered the barley offering. The Israelites were not farmers. They did not plant or harvest. They were nomads, moving from one place to another in tents. Thus barley was rare and had to be purchased so it was quite expensive. Thus the husband would have to be quite serious about his suspicion to shell out the bucks for 3 1/2 lb. of barley. Likewise the wife had to grind the barley to a fine powder. This took some time giving her ample time to consider telling her husband the truth.

Notwithstanding the absence of witnesses, the priest could always use the Urim or Thummim to determine guilt or innocence just like Joshua (Joshua 7:10-25). Despite the priest having the Urim and Thummim at the ready in his breastplate (Ex. 28:13-30, Lev. 8:8), the Law of Jealousy did not provide this remedy to determine the woman’s guilt or innocence. This is an interesting point, either the Urim or Thummin were not consulted because the woman was already determined to be innocent by the lack of witnesses or it was decided to invent a new device to determine the woman’s guilt. Many Christians have wide acceptance of the invention of a new test for guilt and innocence but the Urim and Thummim were continually consulted afterwards for hundreds of years and no other new tests were ever invented.

If indeed the lack of witnesses assured the woman’s innocence, what was the priest to do? The priest administered a test. This test was not a magic potion to determine the innocence or guilt of the woman; the lack of witnesses exonerated her.

Here it is instructive to note that the whole issue is the man’s suspicion that his wife is impure (defiled) and her impurity is undetected (Num. 5: 13,14). The hidden sex act and her impurity are two separate issues since the Priest distinguished between going astray and making herself impure (Num. 5:20,21,29). Indeed the resulting miscarriage after drinking the bitter water is considered to be a result of being both impure and being unfaithful (Num. 5: 27).

Westerners do not see it this way. The sex act with a man that is not her husband makes the woman impure. We think this makes the woman defiled or polluted. No so with the ancient Hebrews. This is going astray. The impurity arises from the resulting pregnancy of the extramarital affair. It is the defiling or polluting of the family genes comprising the new baby that makes her impure.
It is not easy to understand such a Hebrew culture where keeping track of who’s kids were who’s was such an important matter but it was. The best examples of this concept in the Bible are the stories of Ruth and Tamar. In Hebrew culture, the wife had an absolute right to bear children from her husband’s gene pool. In the Bible the mother-in-law, Naomi, lost her husband and two sons. She released her daughters-in-law because she was old, had no husband or other sons to give them, and she could not have sons in the future to father children for them (Ruth 1:3-13). The extent that the community went through to protect the wife’s right sometimes bordered on the bizarre. Tamar lost her husband and so his brother, Onan, was obligated to give her children. He slept with her but refused to get her pregnant so she waited for the youngest brother, Shelah, to grow up to be old enough to give her children. After considerable time she dressed up like a prostitute to trick her father-in-law into sleeping with her (Gen. 38:6-27). She ended up with twin sons. Odd stories, but they point to the importance of Hebrew women having a right to have children with their husbands, or in the case the husband died, the brother, or father of the same family. Any other man would not do. It had to be from the same family. And the resulting children became inheritors of the dead man’s estate. Fathers likewise were concerned about who’s kids they supported since they were required to give double portion of inheritance to the eldest son and regular portions to their other sons (Deut. 21:15-17) and dowry to daughters (Judges 1:12-15).

The priest could not determine whether the wife had gone astray but he could administer a test to ensure that, at least, the husband would not have to support a child that was the result of an extramarital affair.

If the woman agreed (Num. 5:22), she would drink a liquid given to her by the priest (Num. 5:24).

The liquid was composed of holy water in an earthen (or brazen) vessel, dust from the floor of the tabernacle (Num. 5:17), and ink from the words of the curse written on the scrolls (Num. 5:23). The water was bitter and brought the curse of miscarriage (Num. 5:18).

Earthen vessels in contact with meat from sacrificial services or anything unclean were always broken after use (Lev. 6:28, Lev. 15:12). Despite the widespread use of Holy Water in modern Christian traditions, Numbers 5 is the only place in the Bible where Holy Water is mentioned. The word denotes that it could have been sanctified water, consecrated water, or just water that was set aside but it was not the same as regular drinking water. Although not described in Numbers 5, it is likely that the earthen vessel was broken afterwards since it contained an unclean liquid.

If the woman was pregnant the liquid provided by the Priest likely contained an abortifacient. Abortifacients induce miscarriage. Abortifacients come from plant or mineral sources and have been used since ancient times. Greek playwright Aristophanes in 421 BC described use of pennyroyal to terminate pregnacy. An example of a modern day abortifacient is the morning after pill, RU-486.

If the women was pregnant, the abortifacient caused her to discharge the fetus (Num. 5:27). If she was not pregnant, nothing would happen, she would be exonerated (Num.5:28) and her husband was obligated to resume his husbandly responsibility to sleep with her and provide her children (Num.5:29) as was her right (Ruth 1: 10-11; Gen. 38:1-11).

The Law of Jealousy teaches three concepts: 1) The Laws of Moses ensured women had the right to due process in legal matters. 2) The woman had a choice whether to drink the abortifacient and abort the fetus or not. 3) Women, Men, Priests, and the community recognized that, besides the health of the women, personal, and societal considerations were factored in to decide whether to end a pregnancy.

Note: Many Pro-life Christians believe this interpretation of Numbers 5 to be heresy and completely contrary to the teachings of the Bible. They cite verses like Ps. 139:13-16 where God acknowledges knowing his chosen followers before birth (Jer. 1:4-5) and such verses are certainly true. Notwithstanding these verses, Numbers 5 makes no exception if the unfaithful woman is pregnant. Also there are several citations in the Bible where God commands the Israelites to destroy entire civilizations including women and children (Deut. 20:16-18). That God sanctioned the law of jealousy and commanded infanticide is contrary to Western morality. We believe that adults are responsible for evil but children are innocent. The Amorites were evil, no doubt (Gen. 15:13-21). I have listened to ministers inculcate persuasive reasons why Hittite and Amorite children must be slain. Some have said that the children would grow up and then retaliate against Israel. The story of Haman in the book of Esther where he tries to complete the vow of his Amalakite descendants (Deut. 17:16; Deut. 25:17-19; 1 Sam. 15:1-3) to destroy Israel, doesn’t dissuade us from the moral imperative to spare the children and attempt to assimilate them into our culture and values and these explanations are not satisfying. Children are innocent. I can’t reconcile these discrepancies between the moral God we Westerners imagine and the recorded history of the Bible. I have to acknowledge that God has, for whatever reason, sanctioned the killing of innocent children and babies. I am not God’s judge (Isa.40:13,14, 25,26), my thoughts are not his thoughts (Isa. 55:8-11), and his understandings are beyond my abilities to fathom (Isa. 40:28-31).

It's important to look at this in further context--look at some of the verses following the portion you show:

Number 5:21,27-28 NIV ...may the Lord cause you to become a curse among your people when he makes your womb miscarry and your abdomen swell ... And when he has made her drink the water, then, if she has defiled herself and has broken faith with her husband, the water that brings the curse shall enter into her and cause bitter pain, and her womb shall swell, and her thigh shall fall away, and the woman shall become a curse among her people. But if the woman has not defiled herself and is clean, then she shall be free and shall conceive children.

The surrounding context confirms that this is indeed about ending the lives of children. Whether it's meant to miscarry the children currently in her womb or it's meant to be a lifelong curse of miscarrying any potential children (the structure makes it unclear), the fact remains that this is an intentional miscarrying of children for this woman's womb, and I think most everyone would consider that an abortion.

So then the real questions come: Does god condone abortions? Does God not care about babies? Should unborn children be denied rights as human beings?

The resounding answer that this passage gives to all three questions is a powerful NO! This passage shows just how serious God takes the covenant of marriage and how shameful and iniquitous it is when a baby is aborted.

Numbers 5:31 ESV The man shall be free from iniquity, but the woman shall bear her iniquity.

Numbers 5:38 NIV ...and the woman shall become a curse among her people...

This law was meant to be for a woman who intentionally cheated on her husband. As a punishment for this crime her baby, along with any potential children, was to be lost. But it's not God or the priest who aborts the baby, but it's the woman because of her wrongdoing.

This curse on the woman was to be seen as a de-facto abortion. Her iniquity of cheating on her husband would place on herself the iniquity of always miscarrying her children, and for this she would be seen as a curse among her people.

That passage doesn't even say the woman was pregnant, yet pro-aborts use it to rationalize abortion! It shows how desperate they are -- as if they actually cared about what the Book of Numbers said.

When people use that argument, just say, "You are welcome to show me in the Book of Numbers where it says the woman is pregnant (not there). Then you could show where she has an abortion (not there). Then you could show where God taking a life means that we can also take lives in the another fashion for any reason, including those of our own children (not there)."

It is not about abortion, it's about Jealousy and the Lord acting as a witness in trials that otherwise had none.

Note that the NIV is the only translation to use the word 'miscarriage' (see Numbers 5:22 in parallel to 18 other translations). The translators interpret 'Your thigh to rot' and 'to rot [your] thigh' as 'miscarriage'. Whereas the Hebrew for 'thigh', יָרֵ֑ךְ (yā·rêḵ), is translated elsewhere in the NIV Bible as 'side'.

"Gird your sword on your side, you mighty one; clothe yourself with splendor and majesty." (Psalm 45:3 NIV)

And the Hebrew for 'to rot', or 'waste away', נֹפֶ֥לֶת (nō·p̄e·leṯ), is translated elsewhere in the NIV as 'to fall down' (see Judges 19:27 NIV) Similarly, only a couple of translations even translate בֶּ֖טֶן (be·ṭen) and בִּטְנֵ֖ךְ (biṭ·nêḵ) as 'womb'. The rest use 'belly' or 'abdomen'.

Interpreting these scriptures to mean abortion is isolated to the questionable translation given in the NIV.

The section copied below does a really good job of explaining these verses (source link included).

Numbers 5:11–31. The Trial of Jealousy

This law for determining the guilt or innocence of an adulterer is puzzling in many respects. At first it seems heavily biased against the woman for there is no similar requirement for the man. A close examination of the law will show what was involved in it and why the Lord revealed it.

“The rabbins who have commented on this text give us the following information: When any man, prompted by the spirit of jealousy, suspected his wife to have committed adultery, he brought her first before the judges, and accused her of the crime; but as she asserted her innocency, and refused to acknowledge herself guilty, and as he had no witnesses to produce, he required that she be sentenced to drink the waters of bitterness which the law had appointed; that God, by this means, might discover what she wished to conceal. After the judges had heard the accusation and the denial, the man and his wife were both sent to Jerusalem, to appear before the Sanhedrin, who were the sole judges in such matters. The rabbins say that the judges of the Sanhedrin, at first endeavoured with threatenings to confound the woman, and cause her to confess her crime; when she still persisted in her innocence, she was led to the eastern gate of the court of Israel, where she was stripped of the clothes she wore, and dressed in black before a number of persons of her own sex. The priest then told her that if she knew herself to be innocent she had no evil to apprehend; but if she were guilty, she might expect to suffer all that the law threatened; to which she answered, Amen, amen.

“The priest then wrote the words of the law upon a piece of vellum, with ink that had no vitriol in it, that it might be the more easily blotted out. The words written on the vellum were, according to the rabbins, the following:—‘If a strange man have not come near thee, and thou art not polluted by forsaking the bed of thy husband, these bitter waters which I have cursed will not hurt thee: but if thou have gone astray from thy husband, and have polluted thyself by coming near to another man, may thou be accursed of the Lord, and become an example for all his people; may thy thigh rot, and thy belly swell till it burst! may these cursed waters enter into thy belly, and, being swelled therewith, may thy thigh putrefy!’

“After this the priest took a new pitcher, filled it with water out of the brazen bason that was near the altar of burnt-offering, cast some dust into it taken from the pavement of the temple, mingled something bitter, as wormwood, with it, and having read the curses above mentioned to the woman, and received her answer of Amen, he scraped off the curses from the vellum into the pitcher of water. During this time another priest tore her clothes as low as her bosom, made her head bare, untied the tresses of her hair, fastened her torn clothes with a girdle below her breasts, and presented her with the tenth part of an ephah, or about three pints of barley-meal, which was in a frying pan, without oil or incense.

“The other priest, who had prepared the waters of jealousy, then gave them to be drank by the accused person, and as soon as she had swallowed them, he put the pan with the meal in it into her hand. This was waved before the Lord, and a part of it thrown into the fire of the altar. If the woman was innocent, she returned with her husband; and the waters, instead of incommoding her, made her more healthy and fruitful than ever: if on the contrary she were guilty, she was seen immediately to grow pale, her eyes started out of her head, and, lest the temple should be defiled with her death, she was carried out, and died instantly with all the ignominious circumstances related in the curses.” (Clarke, Bible Commentary, 1:634.)

Several points should be noted.

1. Although this ritual focused on the woman, it in no way implied that men who committed adultery were to be excused, for the law clearly stated that adulterers of both sexes were to be stoned (see Leviticus 20:10).

2. In a way, the law provided protection of two different kinds for a woman. First, without this law it is possible that a husband could unjustly accuse his wife of infidelity. If his word alone were sufficient to convict her, she would be in a terrible state indeed. Putting the determination of guilt or innocence into the hands of God rather than into the hands of her husband, or even other men, ensured that she could vindicate herself if she were innocent. The second positive benefit is more subtle but probably is of even greater value. If a husband suspected his wife of adultery, one result would be a terrible strain in the husband-wife relationship. In today’s legal system, with no witnesses to prove her guilt, the court would probably declare her not guilty. But the basis for her acquittal would be a lack of positive evidence of her guilt rather than proof of her innocence. Such a legal declaration, therefore, would do little to alleviate the doubts of the husband and the estrangement would likely continue. Neighbors and friends also would probably harbor lingering suspicions about her innocence. With the trial of jealousy, however, dramatic proof of God’s declaration of her innocence would be irrefutable. The reputation of the woman would be saved and a marriage relationship healed. Thus, true justice and mercy were assured, and the whole matter would be laid promptly to rest.

3. Those who ask why there was no parallel test a woman could ask of her husband should remember that if the accused woman refused to undergo the trial by drinking the water, her action was considered a confession of guilt. Thus, she and her partner in the evil act would be put to death (see Leviticus 20:10). If she attempted to lie and pass the test, but brought the curses upon herself, this result too was considered proof of the guilt of her male partner. It is possible that a wife who believed her husband guilty of infidelity could ask that his suspected partner be put to the trial of jealousy. The outcome would immediately establish the guilt or innocence of her husband as well as that of the other woman. Thus, in a world where the rights of women were often abused, the Lord provided a means for protecting their rights as well as seeing that evil was put away and justice done.

Thus, in a world where the rights of women were often abused, the Lord provided a means for protecting their rights as well as seeing that evil was put away and justice done.

Source: Old Testament Student Manual Genesis-2 Samuel, (1980), 196–202: Numbers 1–12: Wilderness Wanderings, Part 1, (section 17-10: Numbers 5:11–31. The Trial of Jealousy)