George Washington corresponded with many churches, the strong majority of which were orthodox Trinitarian Christian, and he invariably praised them with polite words. Peter A. Lillback argues this as strong evidence that Washington himself was orthodox Christian. However, the larger picture viewed in overall context casts doubt on such notion. The theistic rationalists supported "Christianity" because they supported "religion" in general. Ben Franklin, for instance, was involved in the building of churches which, I think, were used solely by orthodox Trinitarian Christians (like Franklin's friend, the uber-orthodox Christian, George Whitefield). However, Franklin was so creedally indifferent that he asserted it would be entirely appropriate for "the Mufti of Constantinople...to send a missionary to preach Mohammedanism" in those churches.
Likewise, George Washington's praise for the Christians with whom he often corresponded was not limited to orthodox Trinitarian Christians, but was done with the same cavalier creedal indifference which Franklin displayed. In short, Washington and the other theistic rationalists praised a variety of "religions" which preached incompatible claims, as "sound."
Read Washington's words carefully; when he praises the Christian Churches, he never so much insinuates that one must be a Christian to be saved, but only that Christianity makes men moral. And as Franklin once said, if the "ends" (virtue) are achieved, the "means" (which religion you believe in) really don't matter. As such, they believed all religions about which they were aware taught the morality necessary to support republican government and were valid ways to God. So ultimately, what theistic rationalists like Washington valued in Christianity and all other religions was the existence of an overridding Providence, the obligations of moral virtue, and the doctrine of a future state of rewards and punishments.
The key Founders believed Christianity, Judaism, Deism, Unitarianism, Hinduism, Islam, Native American Spirituality, and Pagan-Greco-Roman spirituality all taught the "theistic" minimum that made religion "sound." Things like the Trinity, Incarnation, and Atonement were at best harmless irrationalities, at worst, corruptions which detracted from Christianity's "purity" or "simplicity." Christianity might be "better" than the other religions, but only because of Jesus of Nazareth's superior moral teachings, not because of His claim as second person in the Godhead or His atoning death. The key Founders' creed could be viewed as both a broader form of Deism and a more liberal, rationalistic form of Christianity. Or, it could be viewed as neither strict Deism, nor orthodox Christianity.
Washington and Madison are tricky because whereas Jefferson, Adams, and Franklin leave no doubt in their writings that they believed what I've just described, Washington and Madison invariably spoke in generic terms about God and religion and left some gaps that require detective work to fill in.
So when I see Washington give an infidel church the same praise that he gives orthodox Churches (indeed, the orthodox ministers, with their strict theological teachings are the ones who term such non-orthodoxy, "infidelity") to me this is just one more piece of evidence that strongly points towards Washington being a theistic rationalist like Franklin, Adams, and Jefferson, not an orthodox Christian.
The "infidel" church Washington praised was the Universalist Church in Philadelphia. This Church was Trinitarian -- it was "orthodox" in its Christology -- however, it denied eternal damnation, believing all would be saved. Benjamin Rush eventually converted to such Trinitarian Universalism. Orthodox Christians commonly termed deists "infidels." However, they also termed unitarians, and universalists "infidels" as well (unitarianism and universalism are both elements of theistic rationalism.)
For instance, Bishop Meade was a very distinguished Episcopal Bishop and scholar in the early 19th Century. In his classic, Old Churches, Ministers and Families of Virginia, Bishop Meade warns of "infidelity" being preached from the pulpit of Christian Churches in Virginia. He notes: "I have other reasons for knowing that infidelity, under the specious garb of Universalism, was then finding its way into the pulpit." He then names a few notable Virginia politicians who were on the legislature's Committee on Religion, who supposedly fought the good fight against "infidelity." [Eventually, "infidels" Jefferson and Madison served on that Committee where they introduced Jefferson's historic Virginia Bill on Religious Freedom which separated Church & State there.] George Washington also served on such committee.
As Peter Lillback interprets Meade's passage, "the names of those openly defending historic Christianity against Deism in Virginia are Washington's fellow members of the Committee on Religion." The irony that escapes Lillback is that Washington, in no uncertain terms, praised the very infidelity that Meade condemned. As he wrote to the Universalist Church in Philadelphia:
I thank you cordially for the congratulations, which you offer on my appointment to the office I have the honor to hold in the government of the United States.
It gives me the most sensible pleasure to find, that, in our nation, however different are the sentiments of citizens on religious doctrines, they generally concur in one thing; for their political professions and practices are almost universally friendly to the order and happiness of our civil institutions. I am also happy in finding this disposition particularly evinced by your society. It is, moreover, my earnest desire, that all the members of every association or community, throughout the United States, may make such use of the auspicious years of peace, liberty, and free inquiry, with which they are now favored, as they shall hereafter find occasion to rejoice for having done.
With great satisfaction I embrace this opportunity to express my acknowledgments for the interest my affectionate fellow-citizens have taken in my recovery from a late dangerous indisposition; and I assure you, Gentlemen, that, in mentioning my obligations for the effusions of your benevolent wishes in my behalf, I feel animated with new zeal, that my conduct may ever be worthy of your favorable opinion, as well as such as shall, m every respect, best comport with the character of an intelligent and accountable being.
Washington made it clear that whatever it was he valued about religion, the "infidel" Universalists had it. Further, Washington's line about "however different are the sentiments of citizens on religious doctrines...their political professions and practices are almost universally friendly to the order and happiness of our civil institutions," expresses the radical creedal indifference of theistic rationalism. Not all of these churches were orthodox Trinitarian, even if most of them were. Again, keep in mind the context is he writes this while praising a church whose distinguishing tenet is that it denies eternal damnation and gets called "infidel" for doing so. Washington also defended a Universalist chaplain named John Murray when other chaplains demanded he be forbidden from serving. And Washington also made clear the purpose of the chaplaincy was not to promote the Christian religion to help save men's souls, but rather to accommodate the religious sentiments of the people serving(whatever "religion" they may be) and to take advantage of "religion's" salutary effect on society.