The Puritans put down stakes early on and took the European tack of enforcing theirs as the only allowable official religion in the territories where they lived. In fact, they went beyond their own Anglican roots in England, which by then was accepting presbyterianism and tolerating some other forms of Protestantism. The Puritans turned on Quakers with a vehemence that is Trump-worthy. They accused them of witchcraft, punched holes in their tongues, imprisoned and beat them, forced them out of New England and Virginia, or sent them back to England.
Even before that, French Huguenots had established a colony in what's now Florida, only to be wiped out by the Spanish, who told their king that "they were scattering the odious Lutheran doctrine in these Provinces."
Not until James Madison was there any movement toward tolerance in government policy, and it is Madison's work that allows us to suggest that ours is not necessarily a Christian nation. He and Thomas Jefferson agreed on a plan for establishing religious freedom. Madison wrote later that "religion and Govt. will both exist in greater purity, the less they are mixed together."
That did not stop anti-Catholic fervor, which started with the burning of a convent in Massachusetts in 1834 and built steadily through the growth of the Know-Nothing Party, which of course built anti-Catholic rhetoric into its platform. The Know-Nothings were followed by the Ku Klux Klan. In the 20th century, Al Smith was defeated by anti-Catholics, and even JFK had to reassure the populace that his allegiance was to the nation, not to the Pope. There is even an element of anti-Catholicism in our current immigration hysteria.
Mormonism is an easy religion to mock—the only one I can think of to have inspired its own satirical Broadway show—but the Church of Jesus Christ of Latter Day Saints may be the only religion since the founding that has also inspired its own war. The Utah War of 1857-58 saw the federal government under President Buchanan descend on Utah Territory in an effort to disperse the Mormon population. It was apparently a bloodless war, since the Mormons were underarmed and refused to fire on the soldiers. Mostly they played pranks on the federal troops, blocking their way, burning the grass for their horses, and stampeding their cattle. It's a pretty good David v. Goliath story. But that's not to say that Mormons did not suffer from persecution—their prophet was killed by an armed mob in 1844, and they were forcibly expelled from their homes in the midwest, leading to their re-establishment in Utah Territory under Brigham Young.
And then there's anti-Semitism, both overt and covert. We've had Charles Coughlin and the Christian Front; the Ku Klux Klan, which never met a non-Protestant it couldn't hate; the overt anti-Semitism of Richard Nixon and the slightly quieter but far more dangerous anti-Semitism of FDR, which may have been responsible for the turning away of Jewish refugees and the suppression of information about the Holocaust.
I will say that I can't remember a candidate using anti-religious prejudice as a plank in his campaign—but there does seem to be a bit of a precedent in the Hoover-Smith battle of 1928, where Hoover was careful not to cross the line but let his wife and the Klan spread the message about Smith. In the Nixon-Kennedy battle, Nixon vowed to keep religion out of the campaign while working behind the scenes to build up anti-Catholic fervor. Our local Millard Fillmore was nominated by the Know-Nothings to run a second time for the Presidency in 1856, an election he lost. It was an ironic move, because Fillmore had actually named Brigham Young governor of Utah Territory during his earlier presidency. The only thing he really had in common with nativists was his desire to unite the nation.
Lest we forget, here's the First Amendment:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
It is amazing how the people who tout American exceptionalism, by which I think they mean our unique pursuit of democratic ideals, can be the very same people who are willing to set fire to the Constitution whenever adhering to its principles seems a bit inconvenient. But they have many historical prototypes on which to model their bad behavior.