Is God capable of ever doing wrong?

The answer depends on your definition of "right" and "wrong". Christians base their definitions on God, so everything God does must necessarily be "right" by this definition.

It is not the only definition, however, and many non-Christians will undoubtedly say that some of God's actions, especially some mentioned in the Old Testament are "wrong" according to their definitions.

In some ways, this is a question that makes no sense, simply because God defines what is right and wrong. It's like the questions about an unstoppable object hitting an unmovable object, or asking if God can create a rock that is too heavy for him to lift. The questions themselves are nonsense.

Because God defines what is right and wrong, and that he only does right, asking if God can do wrong is kind of like asking if God can be not God.

All through the Bible, God is the definition of what is right and good, and claims he has the right to make that definition as well. 1 John 1:5 "God is light; in him there is no darkness at all." James 1:17 "Every good and perfect gift is from above, coming down from the Father of the heavenly lights, who does not change like shifting shadows." Job 40:8 "“Would you discredit my justice? Would you condemn me to justify yourself?"

It's still valid to ask the question, because understanding what the question means helps to understand the nature of God.

Catholicism, for one, does not pander to notions of moral relativism, that there is no objective truth and good and evil can't be objectively established. We have no problem taking on hard questions like this, but the answer is a mystery that'll only leave you hanging.

The active will of God is only going to do and has only done the highest good.

• Flooding the earth to cleanse it,
• killing the first born Egyptian sons,
• letting Satan pick on poor old Job and kill his family.

All the things we read in the Bible about God actually doing, not just let happen are indeed good and it's not just the Jewish or Christian conception of good. It's the objective, ordered with the universe conception of good.

• The world was flooded, leaving only Noah and his family alive, because they were actually the only people who had any fear of the Lord in them and if God's plan for salvation was to continue, he was going to have to do something. Everyone else brought judgement upon themselves, they had over a hundred years to repent.
• The first born children of Egypt were killed due to the hardness of Pharaoh's heart, which was caused by God, but He still respects our freewill - and how are the people who allowed the sons of Jacob to become enslaved in the land where their coming was such a blessing not guilty? They had 9 plagues in which to depose Pharaoh which they squandered. In pre-youth worshiping cultures, I think it would be the parents who would be considered the victims here and it's them who were being judged, not their innocent children. God gives life and takes it away, it's not an atrocity for God to be God at the beginning of life any more than it is for Him to be God at the end of your life.
• Job's sons were not living rightly, and even though they didn't do anything to deserve to have their house collapse on them, they could have been doing something instead of partying. Also, that ought to be taken more allegorically anyway - but people often bring it up as God doing something nasty.

Everything else you see or hear about God doing, letting Earthquakes, Tsunamis, Tornadoes happen, these are all things God allows to happen, yeah, but it's always to bring about a greater good. But we can't understand, nor can we convince anyone else with our words.

Christian theology holds that God is holy, righteous, omnipotent, and unchanging (among other things). So, not only is He holy and righteous, but He has all power to maintain His holiness and never change.

All He does is consistent with His character, and His character is holiness and righteousness.

Additionally, God is complete in Himself and has need of nothing, so there's nothing to tempt Him to do what is wrong. We are tempted to do wrong, because we believe that sin will somehow be to our benefit. God would certainly not be so easily deceived.

Joel's answer is a good one, but I would just add that the question is really one that's sort of loaded with certain presupposed authority. For anything to be deemed objectively right or wrong, then that thing needs to be measured against some sort of authoritative transcendent standard (a standard that supercedes the thing in question).

In order for God to do a "wrong," then there would have to be some sort authority or standard higher than God himself against which His actions are measured. Hypothetically, suppose God was deemed to have done something "wrong"...who/what is it that gets to decide or identify it as such?

Christianity presupposes God as the ultimate sovereign and author and sustainer of all things. He creates the cosmos and defines the rules of the game, as it were. There are naunces and consequences of the "rules" that may seem unfortunate to us at times, but there's no higher set of rules that we can point to in order to say they are wrong.

Maybe it's not as related as it seems, but it kind of reminds me of this verse:

Romans 9:21

Has the potter no right over the clay, to make out of the same lump one vessel for honorable use and another for dishonorable use?

As the sole owner of his creations, no one is really in a position to argue that the potter has made the wrong pot.

Christianity defines right and wrong through God. It's not whether God does right or wrong, but rather that something is right because God does it, or is in his nature to do.