When I argue against the idea of a "Christian Nation" it's usually one particular variant (which happens to be the dominant one in religiously conservative circles) that I tackle (some would say have demolished): It's the David Barton, D. James Kennedy, Peter Marshall version that holds "just about all of the Founders were Christians like I am" and the Founders turned to an infallible Bible/orthodox Christian theology as the prime ideological source for American Founding documents. John Locke, their chief ideological influence, was really a "Christian" (meaning an orthodox Christian) who appealed to an infallible Bible for his politics. Indeed, it was really God that "founded" America (the FFs were just His instruments) on "biblical principles." And consequently, as God's agents, they could do little or no wrong.
What I have described might sound like an exaggeration; but those folks posit such an exaggerated and unreal historical narrative. They use a notoriously revisionist "law office" method and defend their case like the late Johnny Cochran defended OJ Simpson.
The shame of it is that there are other "Christian Nation" or "Judeo-Christian" theses that are intellectual and historically defensible, but don't have the same popular appeal as the "Christian Nation" narrative as told by Barton et al.
At American Creation, Kristo Miettinen makes such a case in a fascinating post. But in doing so, by intellectual necessity, he does not argue that the key Founders were "orthodox Trinitarian Christians" OR that they (and their philosophical mentor John Locke) simply appealed to orthodox Christianity/sola scriptura in establishing American Founding principles, especially those that relate to freedom of conscience. No, actually history is more complicated than that.
A few things stand out in Miettinen's post. First, is that he argues John Locke (arguably the most important ideological figure whom the Founders consciously followed) did not hold to an authentically Christian position:
First, many Christian philosophers (e.g. Locke) have held the philosophical view, rather than the Christian view, of conscience. Locke went so far as to take, regarding conscience and its origins, the Freudian view long before Freud did (that conscience is a result of conditioning, holding us back). But apart from Christian philosophers, Christian theologians have has always wrestled with the problem of errors of conscience. The conscience has, in Christian theology, been viewed both as capable of learning on the one hand, and of wandering into error on the other. This is a far cry from the capriciousness of the free will, but it is still indicative of freedom of a sort for the conscience, and unfortunately it has also always formed the basis for Christian persecution: what is free can be influenced, and coercion is just the influential cousin of persuasion.
Indeed, the Founders engaged in historical and philosophical "revisionism" reading things through a Lockean (or a "Whig") lens:
To take another try, many founders quoted Locke and claimed his influence upon the founding. To be sure, they also often cited general “Christian principles”, but still, mustn’t we give Locke (and others) his due? In short, my answer is “no”. The founders were proven historical revisionists, rewriting American history to weave Locke in, ex post facto, where he didn’t belong, e.g. JQ Adams giving a Lockean twist to the pre-Lockean “Mayflower compact” (i.e. Mayflower covenant) thus: “the first example in modern times of a social compact or system of government instituted by voluntary agreement conformable to the laws of nature, by men of equal rights and about to establish their community in a new country.” Given such willingness to interpret pre-Lockean events in Lockean terms, the founders’ enthusiasm for Locke should be seen as rationalization, not explanation.
Finally when it comes to defining "Christianity" Miettinen appeals to a broad understanding of the concept:
To the common claim that the founders weren’t, by and large, Christian, because they were Unitarians, or whatever, my reply is to defend big-tent interpretations of Christianity, at least when we are acting as students of history, rather than engaging in sectarian squabbles. To the historian, the criteria that delimit Christianity must, of course, be different in detail than the criteria that delimit Buddhism, or Marxism, but they should be similarly loose and inclusive. Just as all Marxists are Marxists to anyone but a devout Marxist, so also anyone who attributes historical singularity to Christ (whether as redeemer of mankind, son of a unitary God, one person of a triune God, a member of the divine troika, seal of the prophets, resurrected worker of miracles, etc.) should be acknowledged, by any historian thinking in his capacity as a historian, to be a Christian, even if the historian happens also to be a strict sectarian Christian who denies the true Christianity of competing sects. By this standard, historically speaking, (whatever I think as an orthodox Lutheran), Arians, Nestorians, Mormons, etc., are all Christians for scholarly purposes of historical analysis.
In order to make a “political-theological” connection between American government and "Christianity," one must define Christianity rather liberally to include all sorts of heretical systems that most religiously conservative orthodox Trinitarian Christians find unacceptable. Indeed, those who assert "Mormonism isn't Christianity" are the very ones most likely to assert "American is a Christian Nation." It's only by incorporating such heretical systems as Mormonism etc. into the definition of "Christianity" that America's founding political institutions can be said to rest on a "Christian" foundation. True, Mormons didn't exist during the Founding era (they came later). But America's key Founders disproportionately believed in the Arian and Socinian heresies [like the Mormons and Jehovah's Witnesses, America's key Founders were not orthodox Trinitarians]. Madison was quite clear that civil government had no business whatsoever resolving issues such as whether Arianism, Socinianism, and the like qualified as "Christianity"; indeed this was the secret driver behind his "Memorial and Remonstrance. [And indeed, Madison likely was a theological unitarian himself.]
Besides believing in 1) the unitarian heresies, America's key Founders also believed 2) that the Bible was only partially inspired; 3) that man’s reason (not the Bible) was the ultimate determiner of truth; 4) that most or all religions (including non-biblical ones) were valid ways to God; AND 5) they disbelieved in eternal damnation. If those 5 points can be incorporated into the political understanding of "Christianity" then yes, America can be said to have had an authentically "Christian" Founding.