My friends Gregg Frazer and Tom Van Dyke (and some others) are heatedly debating in the comments section at American Creation how to properly understand American political theology. In a nutshell Gregg has termed American political theology "theistic rationalism," and Tom doesn't like the way that term fails to incorporate "Judeo-Christianity." Gregg's thesis argues "Protestant Christianity" makes up one of three components of "theistic rationalism," (the others being Deism and natural religion) but "theistic rationalism" is still not "Christianity." It is also not "Deism," but something in between Christianity and Deism.
So this post explains how America could be accurately termed a "Protestant Nation," but not necessarily a "Christian Nation," and why it might be right to recognize America's Protestant political theological roots, without terming those roots "Christian."
Terms -- what they mean, and how the user and the listener understand them -- might confuse, if not properly explicated, but rather used in vague generalities. The American Founders, by the way, tended purposefully to not explicate theological terms in their public God talk (and private correspondence with non-theologically like minded figures), because, as non-orthodox theists, they differed, theologically, PROFOUNDLY and irreconcilably with the "orthodox" whose "consent" to their liberal democratic-republican project they needed to procure. Washington for instance, invariably spoke in vague theological terms when he addressed the orthodox ministers who adored him and whom he respected.
And that confusion -- where the "orthodox" think the Founders meant what they wanted them to mean, but didn't -- persists to this day (in large part because of the key Founders' clever way of operating and the hagiographic remake of the FFs by pietist myth-makers like Mason Weems -- figures like David Barton and Peter Marshall are the Parson Weemses of the present era).
For instance, one of the "Christian Nation" crowd's favorite quotations is from John Adams to Thomas Jefferson, 28 June, 1813:
The general principles on which the fathers achieved independence, were...the general principles of Christianity, in which all those sects were united, and the general principles of English and American liberty, in which all those young men united, and which had united all parties in America, in majorities sufficient to assert and maintain her independence.
Now, without understanding the CONTEXT of said quotation one might conclude "Christianity" meant things like infallibility of the Bible, original sin, trinity, incarnation, atonement, eternal damnation. But, one would be wrong as both Jefferson and Adams (especially in that year 1813 when their heterodoxy was most pronounced!) rejected every one of those tenets. And the context of said letter suggests Adams meant clearly some *other* theological system:
Who composed that army of fine young fellows that was then before my eyes? There were among them Roman Catholics, English Episcopalians, Scotch and American Presbyterians, Methodists, Moravians, Anabaptists, German Lutherans, German Calvinists, Universalists, Arians, Priestleyans, Socinians, Independents, Congregationalists, Horse Protestants, and House Protestants,2 Deists and Atheists, and Protestants “qui ne croyent rien.” [Protestants who believe in nothing---JR.] 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.
Not just theological unitarians and universalists, but even deists and atheists, Adams states were united under this "Christian" theological system. What Adams means by "Christianity" here seems not unlike his definition to Samuel Miller, July 8, 1820 where he wrote:
“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.”
In other words, if an atheist was a good person, he was a "Christian." This is not what the followers of David Barton, Peter Marshall and others think when they hear that quote plucked from context. Likewise when they hear or argue America was founded to be a "Christian Nation," "Judeo-Christian Nation," or "Protestant Christian Nation" they likely understand or mean something else.
To them "Protestant Christianity" means Sola-Scriptura, the Bible as the inerrant, infallible word of God, and orthodox doctrines from original sin, to the Trinity to eternal damnation. In this sense Protestant Christianity is that lowest common denominator between Luther and Calvin. [For evidence of this, see the comments section of this post, which includes a comment by John Lofton.]
But this is precisely NOT how America was founded as a "Protestant Nation" in a political-theological sense. Rather, America's Protestant political-theological foundations relate to the following: The word "Protestant" literally means to "protest," or "dissent." As it applies to political-theology, American Protestantism means a private right to religious conscience. In this sense the individual not only has a private right to decide for himself on matters of Trinity and eternal damnation, but also which parts of the Bible are valid.
In this sense, America was founded, politically, to be a "Protestant Nation," with Thomas Jefferson the quintessential American (political) Protestant. Jefferson certainly thought of himself as a "Protestant Christian," and was a lifelong member of the Anglican-Episcopal club. He also devoutly believed in an active personal God. However, his "Protestantism" led him to reject, not just (his words) "[t]he immaculate conception of Jesus, his deification, the creation of the world by him, his miraculous powers, his resurrection and visible ascension, his corporeal presence in the Eucharist, the Trinity; original sin, atonement, regeneration, election, orders of Hierarchy, &c.," but also large parts of the Bible as valid revelation.
So, as Jefferson illustrates, it's possible to be a Protestant, an Anglican, but arguably not a "Christian." One could argue Jefferson's rejection of the infallibility of the Bible did in his "Christianity." But even if we accept Sola-Scriptura, American political Protestantism still might not qualify as "Christian."
Case in point, another monumental influence on the American Founding politics: Rev. Charles Chauncy. As historian and President of Wakeforest University Nathan Hatch described Chauncy:
Charles Chauncy, pastor of Boston’s First Church for sixty years (1727-1787), is the most prominent example of an exclusive appeal to Biblical authority in order to unravel theological orthodoxy. Chauncy was persuaded to emphasize Bible study by reading the works of English divines, such as Samuel Clarke’s The Scripture-Doctrine of the Trinity (London, 1712) and John Taylor’s The Scripture-Doctrine of Original Sin (London, 1740). Both authors used a “free, impartial and diligent” method of examining Scripture to JETTISON, respectively, the doctrines of the Trinity and of Original Sin. 
During the 1750s, after the Great Awakening, Charles Chauncy spent seven years engaged in the approach to Bible study expounded by these English authors. In the spring of 1754 he wrote to a friend,
“I have made the Scriptures my sole study for about two years; and I think I have attained to a clearer understanding of them than I ever had before.”
His studies led him to draft a lengthy manuscript in which he REJECTED the idea of eternal punishment and embraced universalism....He explained to Ezra Stiles, “The whole is written from the Scripture account of the thing and not from any human scheme.” This unorthodox biblicist would have been gratified indeed by the reaction of one minister who, finding the book’s arguments convincing, wrote,
“He has placed many texts and passages of Scripture in a light altogether new to me, and I cannot help thinking his system not only rational, but Scriptural.” 
So as it turns out, America's Protestant political-theological foundations mean a fundamental right to private judgment in matters of religious conscience. This political theology teaches the right to religious judgment is so private and individualized, that matters such as original sin, trinity, eternal damnation, and which parts of the Bible are valid are consigned to the realm of the private conscience and are driven from politics; they play no part in America's political-theological foundation.
The question then remains is such a Protestant system that posits Providence and a special place for Jesus, but refuses to take a position on original sin, the Trinity, Jesus as the only way to God, eternal damnation, and the infallibility of the Bible, "Christian"?
How we answer the question determines whether America's Founding political theology is aptly termed "Christian" or not. If not, we could say America had a "Protestant" but not a "Christian" founding.