Did Argentine President Juan Peron grant free access to the radio, the press and two soccer stadiums to a preacher called Tommy Hicks in 1954?

Short Answer

At least some of the above seems to be true. Academic sources confirm that (1) Theodore 'Tommy' Hicks did meet Peron, (2) the use of at least two stadiums was granted (Atlanta and Huracan, both in Buenos Aires), (3) meetings at these stadium were very well-attended (though numbers are disputed), and (4) there was both press and radio coverage.

It cannot be confirmed that Hicks and the evangelism committee had the "freedom to preach anywhere they wanted". In fact, there is no evidence that they held any meetings outside of Buenos Aires.

Juan Peron's motives for granting these favours were political rather than religious, and occurred at a time when there was much friction between the Peronists and the Catholic Church.

Details

Argentine historian Lila Caimari (University of San Andrés, author of "numerous articles and book chapters on various dimensions of Argentine social and cultural history") mentions Hicks meeting Juan Peron and having use of the Atlanta stadium for several months:

In March 1954, Perón met with Pentecostal pastors Arvizu and Hicks, in an interview widely covered by the official press. This public manifestation of support naturally produced new Catholic complaints, complaints which failed to diminish the goodwill of the government toward Pentecostals when they launched a great Evangelical campaign led by Rev. Hicks.

Source: Lila Caimari, 'Peronist Christianity and Non-Catholic Religions: Politics and Ecumenism (1943–55)' (Canadian Journal of Latin American and Caribbean Studies, Vol. 20, No. 39/40, Special Issue: Cárdenas, Vargas, Perón and the Jews (1995), pp. 105-124).

Hicks' meeting with Peron took place on the 16th of March, 1954 at the Casa Rosada, the office of the Argentine president. Another source, Seth N. Zielicke, presents a not dissimilar story. Although he does find inconsistencies in some accounts of the time, these do not concern the basic facts asked about by the OP. Zielicke, a PhD student at the time of writing, states:

After meeting with Perón, Hicks obtained the necessary permits for the stadiums and received free access to the press and the radio. The next day, evangelicals were greatly surprised to find a picture in the newspaper of Hicks standing next to President Perón with the necessary permits.

Source: Seth N. Zielicke, 'The Role of American Evangelist Tommy Hicks in the Development of Argentine Pentecostalism', in Michael Wilkinson (ed.), 'Global Pentecostal Movements: Migration, Mission, and Public Religion' (2012).

There was clearly a significant degree of official support, which proved to be more than a little controversial:

The activities of this pastor took on the dimensions of a real social phenomenon. Between May and June 1954, Hicks produced daily "miracles" before an enormous crowd at the soccer stadium of Atlanta. All sorts of stories about the "Atlanta magician" began to circulate: he cured the sick, the blind, the deaf and the dumb-and all of this in Christ's name. The government's collaboration with the Hicks campaign was no secret: the preacher was able to obtain a special permit for mass public assemblies - a privilege granted to few in 1954 - and the police limited itself to maintaining order at the meetings.

Source: Lila Caimari

As the meetings drew larger and larger crowds,

On Saturday, May 22, Hicks and the leaders moved the meetings to the Huracán Stadium, the largest in the country, with a contested capacity of 100,000.

Source: Seth N. Zielicke

Zielicke notes disagreements over the sizes of these meetings, with estimates ranging up to 200,000. The president of Huracan football club estimated 180,000, though it's not clear if this happened for one or for several meetings.

Unlike Zielicke, Caimari does not mention of radio coverage, but 'access to radio' is mentioned in the Encyclopedia of Protestantism (along with the alleged curing of Peron's skin disease):

In 1954, evangelist Tommy Hicks appeared to have healed President Juan Peron’s skin condition; Peron gave Hicks access to Argentinean radio and allowed him to rent a large stadium in Buenos Aires. During his two months in Argentina, several hundred thousand attended Hicks’s meetings, and a national Pentecostal movement was created.

Source: J. Gordon Melton (ed.), ‘Encyclopedia of Protestantism’ (2005). Note: Melton is Director of the Institute for the Study of American Religion in Santa Barbara, California.

The above photo is taken from a source which may be considered dubious, but there is no obvious evidence that it has been doctored (and there seems to be little doubt that Hicks did meet Peron - the skin cure claim is, of course, another matter entirely). The same also mentions the use of the Huracan stadium. Source: Argentina Shaken by Unprecedented Revival, July 1954

Hicks' time in Argentina is also mentioned in another academic source, along with the use of a soccer stadium:

Peron had an interview with Tommy Hicks, a Pentecostal faith-healer from the United States, and the government subsequently allowed him to use a soccer stadium for his healing campaigns. This occurred at the same time that Protestants were deprived of having meetings in public places.

Source: David F. D'Amico, 'Religious Liberty in Argentina during the First Perón Regime, 1943-1955', in Church History, Vol. 46, No. 4 (C.U.P., Dec., 1977), pp. 490-503

The mention of restrictions on religious gatherings gives some clue as the background to all this, but the main - and increasingly acrimonious - conflict was between the Peron regime and the Catholic church. Unsurprisingly, the amount of publicity generated by Hicks caused much controversy and angered the Catholic church:

The Catholic reaction was immediate. The "Consortium of Catholic Medical Doctors'' published a letter denouncing the total lack of scientific control over Hicks's "miracles," as well as the irresponsibility demonstrated by the "surprising publicity" surrounding them.

The invitation to Hicks had much more to do with politics than with religion:

Perón's religious conceptions can hardly explain the official support for Pentecostals or Spiritists. In this broad, if superficial, endors*m*nt of religious freedom one may divine intentions somewhat less than spiritual: the government wanted to associate itself with religious practices more openly Peronist than those of the Catholic church, where both allies and opponents - the latter increasing in number each day - could be found attending the same mass.

Source: Caimari

Note:

Hicks meeting Peron is also discussed in: Humberto H. Cucchetti, 'Religión y política en Argentina y en Mendoza (1943 – 1955): lo religioso en el primer peronismo', but there are no details on stadiums etc. (and Hicks is referred to as 'Theodore').