According to chapter 19 of the former, it was Norman Cousins, a political activist and editor of a publication called the Saturday Review, who telephoned the Vatican on 24th October 1962 and suggested that the Pope should make an appeal for peace. Pope John was initially concerned that what he said might offend the Soviet side, but Cousins assured him that the terms of the appeal could be checked with each side before publication. Pope John is reported to have stayed up until midnight preparing a radio message that he delivered the next day, 25th October 1962. Given the genuine fear that nuclear war was a possible outcome of the crisis, Pope John's words are very powerful. Appealing to the consciences of those who held power, he said:
May they hear the anguished cry which rises to heaven from every corner of the earth, from innocent children to old men, from persons and communities: peace, peace!
Vatican Radio's coverage marking Pope John's radio broadcast is here, and gives an account suggesting that the initial approach suggesting the Pope issue an appeal for peace came from President Kennedy.
According to Meriol Trevor, Russian premier Kruschev acknowledged to Norman Cousins that he had been moved by the Pope's appeal, which contributed to the easing of tension. The Vatican Radio coverage suggests that the broadcast, while not a crucial factor in bringing the crisis to a conclusion, was nevertheless a factor among others that led to its conclusion.
This article at Crisis Magazine gives an impression of a stronger influence of Pope John's appeal. It also refers indirectly to the part played by John XXIII in helping achieve the nuclear test ban treaty between the Soviets and the United States in 1963.