This dynamic is key to my historical research on the Founding & Christianity. There are many possible meanings of a "Christian nation" some valid and some not. The meaning posited by the "Christian America" proponents defines "Christianity" and "a Christian" fairly tightly according to its orthodox biblical ideas. They really do believe that "God" (meaning their specific understanding of God) founded America and as such He would use "real Christians" in a born-again, regenerate sense to do his bidding. They are probably already confounded by the fact that God would raise up John Adams and Thomas Jefferson, who clearly were not "Christians" in this sense (though they understood themselves to be "Christians" not "Deists") to be second and third Presidents (many of them are ignorant of Adams' heterodoxy, however). And they desperately try to latch on to whatever thin evidence there is that Washington and Madison were "real Christians." If the first four (or five) Presidents from Washington to Monroe were indeed non-Trinitarians who rejected eternal damnation, salvation through Christ alone, and probably didn't believe the Bible the inerrant, infallible Word of God, why would God raise up such leaders to "Found" America, His nation?
I'll again stress that the key Founders who believed what I put in bold did NOT tend to consider themselves "Deists" but "Christians." This was the kind of "Christianity" that drove the American Founding from the top (not necessarily from the bottom), if it's fair to even call it "Christianity."
The key Founders, in a sense, had to call it "Christianity"; if they called themselves Deists and rejected "Christianity" out of hand, the orthodox easier would have been able to separate themselves from this religious system; but, were that to happen, the different sects couldn't have come together to "Found" America. Instead, the key Founders had to convince the orthodox that the "American project" really did derive from "Christianity," properly understood. [Note: I am NOT arguing the key Founders were secret Deists; from reading their letters, I conclude they really were secret unitarians who thought of their system as a purified, rational form of "Christianity."] And they left quotations to be taken out of context by folks who wish for an orthodox Christian Founding. For instance, one of Christian America proponents favorite proof quotes, from John Adams to Thomas Jefferson, June 28, 1813:
The general principles, on which the Fathers Atchieved [sic] Independence, were…the general Principles of Christianity, in which all those Sects were United: and the general Principles of English and American Liberty…Now I will avow, that I then believed, and now believe, that those general Principles of Christianity, are as eternal and immutable, as the Existence and Attributes of God; and that those Principles of Liberty, are as unalterable as human Nature and our terrestrial, mundane System.
The problem is what Adams meant by "general principles of Christianity." The orthodox reading such a quotation might HOPE that Adams meant such things as original sin, the trinity, incarnation, atonement, infallibility of the Bible, regeneration, eternal damnation. But alas, he didn't. In fact, Adams and Jefferson rejected every single one of those tenets. And elsewhere in that very letter Adams explains exactly what he meant by "general principles of Christianity."
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.” As he 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.
To conclude, if we define "Christianity" broadly to include the heterodox ideas of Founding era figures, Adams and Jefferson, and from today, Obama [and many other liberal, moderate, and conservative politicians], then yes, America could be said to have had a "Christian" Founding and still is a "Christian nation."
The problem is that's not how proponents of the "Christian America" thesis understand "Christianity." They don't want what they regard as false, heretical and nominally Christian teachings to have *any* place in the definition of "Christianity," certainly not in the "Christian" that qualifies "Nation" when they argue America was founded to be a "Christian Nation."