Can something qualify as "Protestant" but not "Christian"? Arguably yes. I see America's Founding as a "Protestant" event, but not necessarily a "Christian" one, if Christianity defines according to its Trinitarian orthodoxy as evangelicals and Catholics believe. If Christianity defines in a broader, more theologically liberal sense, then I can see America as founded on Protestant Christian theology. Ben Franklin and Thomas Jefferson were, in my opinion, more likely to identify as or understand themselves "Protestant Christians," not Deists. To Jefferson such "Protestant Christianity" rejected the following:
The 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.
The million dollar question is can such a system that rejects "[t]he immaculate conception of Jesus, his deification, the creation of the world by him, his miraculous powers, his resurrection and visible ascension...the Trinity; original sin, atonement, regeneration...," aptly be termed "Christian." I put ellipses under two tenets of Catholicism, and one of Calvinism, that aren't necessarily part of orthodox Christianity. The rest of what Jefferson rejected are non-negotiable tenets of orthodox Christianity.
Yet, Jefferson was being very "Protestant" in doing so. Protestantism means to protest or dissent, not to be ruled by Church hierarchy or creeds. It was in this respect that Protestantism paved the way for religious and political liberty, enlightenment, and the consequent rejection of Christian orthodoxy. The latter is a point that Roman Catholic critics of Protestantism have long made. Even though evangelicals and Catholics agree on Trinitarian orthodoxy, by putting the Bible and the proper way to understand the Christian religion in the hands of individual ministers and believers themselves, (as opposed to Church hierarchy) what stops the individual from jettisoning Trinitarian orthodoxy?
It could be sola-scriptura. However, 1) some liberal Protestants like America's key Founders rejected sola-scriptura. And 2) other liberal Protestants of the Founding era used sola-scriptura to reject Trinitarian orthodoxy. In The Democratization of American Christianity, Nathan Hatch describes Charles Chauncy, one of the most influential preachers who argued on behalf of the American Revolution:
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.
Evangelical promoters of the Protestant Christian America thesis are fond of noting (with some truth) that American Protestants had serious problems with Roman Catholicism and when they complained about ecclesiastical authorities it was often done in the context of disagreeing with certain elements of Roman Catholicism. What they fail to point out is that this anti-hierarchical, anti-creedal "Protestant" mindset led many of them to associate not just Calvinism but orthodox Trinitarian Christianity itself with Romanism or Popishness. For instance, when I emailed Ellis Sandoz and asked him to comment on John Adams' rejection of the Trinity where Adams states the following:
"The Trinity was carried in a general council by one vote against a quaternity; the Virgin Mary lost an equality with the Father, Son, and Spirit only by a single suffrage."
-- John Adams to Benjamin Rush, June 12, 1812.
"If I understand the Doctrine, it is, that if God the first second or third or all three together are united with or in a Man, the whole Animal becomes a God and his Mother is the Mother of God.
"It grieves me: it shocks me to write in this stile upon a subject the most adorable that any finite Intelligence can contemplate or embrace: but if ever Mankind are to be superior to the Brutes, sacerdotal Impostures must be exposed."
-- John Adams to Francis van der Kemp, October 23, 1816.
Sandoz noted Adams' tone seemed anti-Marian, and anti-Roman Catholic. And indeed, Adams was arguably an anti-Roman Catholic bigot. Note though that Adams' anti-Catholicism led him to associate the Trinity and Nicene Creed with the false teachings of the Roman Church.
Likewise, Ben Franklin, who made the following argument under the auspices of "Protestant Christianity" not Deism, associates not just Roman Catholicism but Presbyterianism/Calvinism with "Priests" who introduced erroneous doctrines into Christianity.
But lest they shou’d imagine that one of their strongest Objections hinted at here, and elsewhere, is designedly overlook’d, as being unanswerable, viz. our lost and undone State by Nature, as it is commonly call’d, proceeding undoubtedly from the Imputation of old Father Adam’s first Guilt. To this I answer once for all, that I look upon this Opinion every whit as ridiculous as that of Imputed Righteousness. ’Tis a Notion invented, a Bugbear set up by Priests (whether Popish or Presbyterian I know not) to fright and scare an unthinking Populace out of their Senses, and inspire them with Terror, to answer the little selfish Ends of the Inventors and Propagators. ’Tis absurd in it self, and therefore cannot be father’d upon the Christian Religion as deliver’d in the Gospel. Moral Guilt is so personal a Thing, that it cannot possibly in the Nature of Things be transferr’d from one Man to Myriads of others, that were no way accessary to it. And to suppose a Man liable to Punishment upon account of the Guilt of another, is unreasonable; and actually to punish him for it, is unjust and cruel.
-- A Defense of Mr. Hemphill’s Observations, 1735.
Note Franklin's argument: True Christianity rejects original sin, a doctrine that "Priests" either Popish or Presbyterian introduced into Christianity. Franklin's conflation of Calvinism with Roman Catholicism reminds me of Philip Pulman's alternate universe where John Calvin was Pope of the Roman Catholic Church in Geneva.
Finally, a note on how George Washington fits into this. Brian Tubbs does a good job explaining Peter Lillback's chapter on George Washington and Anglicanism/Episcopalianism. In short, the Anglican Church or "Church of England" was, obviously, officially against the Revolution; the high-church Anglicans -- those devoted to hierarchy -- were more likely to be loyalists, those who rebelled tended to be "low-church" Anglicans. Here is how Tubbs, following Lillback, describes the split:
High Church Anglicanism was philosophically in agreement with the doctrine of apostolic succession, as embraced by the Roman Catholic Church. The difference, of course, being that the King of England and the Archbishop of Canterbury spoke with apostolic authority for the day -- as opposed to the Vatican in Rome.
The influence of the Protestant Reformation in England, however, had divided the Anglican Communion into two groups over this doctrine. High Church Anglicans favored strict apostolic succession, whereas Low Church Anglicans adhered more toward scriptural authority.
They conclude Washington was an orthodox Trinitarian of the low-church Anglican variety. And they support this conclusion by showing Washington's (obvious) problems he would have with the high-churchers who remained loyalists. That is why, they believe, Washington avoided communion in the Church.
The problem with this analysis is that it offers a false dichotomy between the orthodox Trinitarian, scripturally based, high-church v. orthodox Trinitarian, scripturally based, low-church Anglicanism. Indeed, it tries to make low-church Anglicanism as more scripturally based, less infected with Roman Catholic elements. In his book, Lillback describes low-church Anglicanism as "Calvinistic." Their description of high-church Anglicanism is more or less accurate. And Calvinistic, scripturally based thought was certainly a movement within low-church Anglicanism. However, so was latitudinarianism, which defines as creedal indifference. Lillback, to his credit, notes latitudinarianism as part of the low-church movement, and Washington as a latitudinarian. He does not, however, adequately explain why "latitudinarianism" must remain true to orthodox Trinitarianism. Indeed, Calvinism is not "latitudinarian," but arguably the antithesis of it. These were two separate lines of thought in low-church Anglicanism.
This is especially apt given what we've seen above, that many notable "Protestants" used their freedom from hierarchy to think for themselves and reject Trinitarian orthodoxy. And indeed, many of those "low-church Anglicans," latitudinarian as they were, often slipped into such creedal indifference that their religion became deistic or unitarian. Indeed these nominal Anglicans whose beliefs were deistic-unitarian include Washington's fellow Founding Fathers and notable Virginians like Jefferson and Madison as well as G. Morris, Franklin, and others. David L. Holmes of William and Mary, one of the most distinguished scholars of American religion and the Anglican/Episcopal Church, writes about this dynamic in his book, A Brief History of the Episcopal Church.
Lillback's only smoking gun proving orthodox Trinitarianism is GW took "oaths" to such when becoming a Godfather and Vestryman. But, from my reading of those oaths, Washington took an oath not simply to orthodox Trinitarianism, but HIGH CHURCH Anglicanism, or the "Doctrine and Discipline of the Church of England, as by law established." And both his avoidance of communion and arguably his act of political rebellion against England itself violated those very oaths.