Of course, you can read Kennedy's speech for yourself, and you'll see that he says nothing about denying people of faith any role in public affairs. He specifically says that people of all faiths should be able to participate in politics:
I believe in an America where religious intolerance will someday end; where all men and all churches are treated as equal; where every man has the same right to attend or not attend the church of his choice; where there is no Catholic vote, no anti-Catholic vote, no bloc voting of any kind; and where Catholics, Protestants and Jews, at both the lay and pastoral level, will refrain from those attitudes of disdain and division which have so often marred their works in the past, and promote instead the American ideal of brotherhood.That hardly seems like something that would cause someone to throw up.
So what so offended Santorum?
I think it was this later section:
That is what made Santorum want to throw up.
Whatever issue may come before me as president — on birth control, divorce, censorship, gambling or any other subject — I will make my decision in accordance with these views, in accordance with what my conscience tells me to be the national interest, and without regard to outside religious pressures or dictates. And no power or threat of punishment could cause me to decide otherwise.
But if the time should ever come — and I do not concede any conflict to be even remotely possible — when my office would require me to either violate my conscience or violate the national interest, then I would resign the office; and I hope any conscientious public servant would do the same.
The goal of Kennedy's speech was to assure the public in general, and Protestants in particular, that as President he would not try to impose the teachings of the Catholic Church on Americans. And the paragraphs quoted above are the most forceful expression of that goal.
And that is what Santorum disagrees with. Santorum has very strong opinions on birth control and abortion and divorce, and Santorum believes that his views on birth control and abortion and just about everything else are superior to all other views. If elected President, it would be not just his right, but his duty to impose those views on all other Americans.
Which makes me want to throw up.