I just read a neat article by one William G. McLoughlin, late professor at Brown University. It was in "Essays on The American Revolution" edited by Stephen G. Kurtz, and James H. Hutson.
The article is probably the best scholarly claim I've ever read of the kernel of truth inherent in the "Christan Nation" claim. The article understands that the ideas and ideals of the American Founding (i.e., the Declaration of Independence, the Constitution, the Federalist Papers, the notion of inalienable rights) have little to do with traditional biblical Christianity. Yet, it stresses that under the original Constitution, religion was left to the states and vestiges of the old "Christian commonwealths" persisted at the state level for some time. Not in Virginia, at least not after 1786, when Jefferson's historic Statute on Religious Freedom became the law of the land, and never in Rhode Island which was Roger Williams' baby (Williams thought the idea of a "Christian Commonwealth" to be blasphemous; and that got him banished the John Winthrop's "Christian Commonwealth" in Puritan Massachusetts). But many/most states post 1789 had laws that favored or otherwise established "Protestant Christianity" in some sense.
The article tellingly asks: "How did the universal spirit of the rights of man become in the end a new national establishment that excluded non-Protestants from full religious equality?" [p. 209.]
That question exemplifies the nuanced dynamic that we need to appreciate. America was "founded" according to a certain set of ideals (see the Declaration of Independence). And those ideals that hold all men are created equal and endowed with unalienable rights also demand "full religious equality" be granted to, as Jefferson put it, "the Jew and the Gentile, the Christian and Mahometan, the Hindoo, and Infidel of every denomination."
Yet, in 1811 in the State of New York, Chancellor Kent in "People v. Ruggles" could uphold a common law blasphemy conviction and state:
Nor are we bound, by any expressions in the constitution, as some have strangely supposed, either not to punish at all, or to punish indiscriminately the like attacks upon the religion of Mahomet or of the grand Lama; and for this plain reason, that the case assumes that we are a christian people, and the morality of the country is deeply ingrafted upon christianity, and not upon the doctrines or worship of those impostors.
Indeed the Courts in some states were still declaring "Christianity" to be part of the common law. And luminaries Joseph Story and John Marshall argued Christianity to have some kind of organic connection to American civil law. The culmination of the "Christian Nation" fervor reached its fever pitch in the Holy Trinity case where the Supreme Court declared American to be a "Christian Nation."
So there is rhetoric and practice from the Founding era and the hagiographic 19th Century that supports the Christian Nation claim. What destroys the Christian Nation claim (as I see it) is that America, though influenced by Christian morality was not founded on its orthodox Christian doctrines. Indeed those who most famously posited "American" principles like Thomas Jefferson, John Adams and Ben Franklin rejected all of Christianity's orthodox doctrines. And others like James Madison and George Washington ignored them. Further as noted above, the ideas contained in the Declaration of independence, Constitution and Federalist Papers have little to do with traditional biblical Christianity.
Why is the word "orthodox" important? To those who most vociferously defend America's public "Christian Heritage," "Christianity" defines according to its orthodox doctrines. Reject those and you reject Christianity and present some "non-Christian" system under the auspices of "Christianity" (as do presently the Mormons, Jehovah's Witnesses and other supposed "cults"; back then instead of terming them "cults" as they do today, the orthodox would simply call such beliefs like those privately posited by America's key Founders, "heresy" at best, "infidelity" at worst).
What do original sin, the trinity, incarnation, atonement, infallibility of the Bible, and eternal damnation have to do with "Christianity"? If the answer is they are "negotiable" and "unimportant" then yes we can say America was founded to be a "Christian Nation." If those doctrines, on the other hand, are central to the Christian faith, then no, America cannot be said to be "Christian" in a public civil sense.
According to America's Founding thought (secretly posited, as it may have been) to be a "Christian" meant to be a "good person." As John Adams put it:
“I believe with Justin Martyr, that all good men are Christians, and I believe there have been, and are, good men in all nations, sincere and conscientious.”
– John Adams to Samuel Miller, July 8, 1820.
Indeed, this is the definition of "Christianity" that Adams used when he stated:
The general Principles, on which the Fathers Atchieved Independence, were...the general Principles of Christianity...[a]nd the general Principles of English and American Liberty....
This is one of the Christian Nation crowd's most oft-repeated quotations. Taken out of context, it seems to support their claim. Reading the rest of the letter, it destroys it. Keep in mind Adams himself was not a "Christian" according to the way the Christian America thesis understands the term. And he said in that same letter Unitarians [Arians, Priestleyans, Socinians], Universalists, Deists and Atheists were all united the "general principles of Christianity" as he understood them:
Who composed that Army of fine young Fellows that was then before my Eyes? There were among them, Roman Catholicks, English Episcopalians, Scotch and American Presbyterians, Methodists, Moravians, Anababtists, German Lutherans, German Calvinists Universalists, Arians, Priestleyans, Socinians, Independents, Congregationalists, Horse Protestants and House Protestants, Deists and Atheists; and "Protestans qui ne croyent rien ["Protestants who believe nothing"]." Very few however of several of these Species. Nevertheless all Educated in the general Principles of Christianity: and the general Principles of English and American Liberty.
How could a Deist or even an "atheist" be "united" with the orthodox on any Christian principles? The answer is simple. According to Adams being a Christian meant being a good person. If an atheist was a good person, he was a "Christian."
So ultimately it was this unitarianism that presented itself under the auspices of "Christianity" that united the sects. But the plan was not for the unitarian ideals to immediately transform American society (such immediate transformation backfired in Revolutionary France), but for the transformation to occur slowly over time. But those changes did occur. American society gradually secularized because of its secret founding principles.
As Thomas Pangle put it, Jefferson's real goal was “conformity based on indifference; not diversity, but the tepid and thoughtless uniformity of Unitarianism in a society where Unitarians no longer have to defend and prove themselves.” This perfectly describes Adams' thoughts as well.