Unlike Jefferson, Adams, and Franklin, whose writings clearly explicate their religious creed, Washington and Madison leave some room for doubt because of their reticence to discuss what they exactly believed. Numerous smoking gun quotations prove that Jefferson, Adams, and Franklin denied the tenets of orthodox Christianity, but not with Washington or Madison. Rather, those two left clues (which I think strongly point towards their believing the same creed as Jefferson, Adams, and Franklin).
For the side arguing Washington's orthodoxy, the smoking gun they argue proves Washington was Christian is that he took oaths to the Trinitarian creeds of the Anglican Church when he became a vestryman and a godfather. If he took oaths while not believing in them, then, the pietists argue, Washington was a hypocrite.
I would respond that those oaths were by in large perfunctory -- a means to an end. If you wanted to become a vestryman (a largely political position in Anglican Virginia) or a Godfather, you had to take those oaths. And Jefferson, who clearly didn't believe in the creeds of the Anglican Church, was also, like Washington, a vestryman. But Jefferson refused to be a godfather and explained he did so because he didn't want to take an oath to the Trinity and its related doctrines in which he didn't believe. Though, as I noted here, the Trinity clearly irritated Jefferson in a way in which it didn't seem to Franklin and Washington. Washington seemed more indifferent or agnostic on the creeds of orthodoxy in which Jefferson actively disbelieved (and indeed, Madison testified that he thought Washington was agnostic on "the arguments for Christianity, and for the different systems of religion" and didn't believe he "in fact...formed definite opinions on the subject").
Here is the exact oath Washington took to become a vestryman:
I, A B, do declare that I will be conformable to the Doctrine and Discipline of the Church of England, as by law established.,
We know that Washington systematically refused to take communion perhaps his entire life, but at the very least from the time of the revolution till his death, well after the Anglican Church became the Protestant Episcopal Church and severed ties with Great Britain. So I was examining "the Doctrine and Discipline of the Church of England, as by law established," -- the 39 Articles of Religion, specifically as it related to the Lord's Supper.
Article XXV: Of the Sacraments
Sacraments ordained of Christ be not only badges or tokens of Christian men's profession, but rather they be certain sure witnesses, and effectual signs of grace, and God's good will towards us, by the which he doth work invisibly in us, and doth not only quicken, but also strengthen and confirm our Faith in him.
There are two Sacraments ordained of Christ our Lord in the Gospel, that is to say, Baptism, and the Supper of the Lord.
Those five commonly called Sacraments, that is to say, Confirmation, Penance, Orders, Matrimony, and extreme Unction, are not to be counted for Sacraments of the Gospel, being such as have grown partly of the corrupt following of the Apostles, partly are states of life allowed in the Scriptures; but yet have not like nature of Sacraments with Baptism, and the Lord's Supper, for that they have not any visible sign or ceremony ordained of God.
The Sacraments were not ordained of Christ to be gazed upon, or to be carried about, but that we should duly use them. And in such only as worthily receive the same have they a wholesome effect or operation: but they that receive them unworthily purchase to themselves damnation, as Saint Paul saith.
Article XXVI: Of the Unworthiness of the Ministers, which hinders not the effect of the Sacrament
Although in the visible Church the evil be ever mingled with the good, and sometimes the evil have chief authority in the Ministration of the Word and Sacraments, yet forasmuch as they do not the same in their own name, but in Christ's, and do minister by his commission and authority, we may use their Ministry, both in hearing the Word of God, and in the receiving of the Sacraments. Neither is the effect of Christ's ordinance taken away by their wickedness, nor the grace of God's gifts diminished from such as by faith and rightly do receive the Sacraments ministered unto them; which be effectual, because of Christ's institution and promise, although they be ministered by evil men.
Nevertheless it appertaineth to the discipline of the Church, that inquiry be made of evil Ministers, and that they be accused by those that have knowledge of their offences; and finally being found guilty, by just judgement be deposed.
Article XXVIII: Of the Lord's Supper
The Supper of the Lord is not only a sign of the love that Christians ought to have among themselves one to another; but rather it is a Sacrament of our Redemption by Christ's death: insomuch that to such as rightly, worthily, and with faith, receive the same, the Bread which we break is a partaking of the Body of Christ; and likewise the Cup of Blessing is a partaking of the Blood of Christ.
Transubstantiation (or the change of the substance of Bread and Wine) in the Supper of the Lord, cannot be proved by holy Writ; but is repugnant to the plain words of Scripture, overthroweth the nature of a Sacrament, and hath given occasion to many superstitions.
The Body of Christ is given, taken, and eaten, in the Supper, only after an heavenly and spiritual manner. And the mean whereby the Body of Christ is received and eaten in the Supper is Faith.
The Sacrament of the Lord's Supper was not by Christ's ordinance reserved, carried about, lifted up, or worshipped.
Article XXIX: Of the Wicked which do not eat the Body of Christ in the use of the Lord's Supper
The Wicked, and such as be void of a lively faith, although they do carnally and visibly press with their teeth (as Saint Augustine saith) the Sacrament of the Body and Blood of Christ, yet in no wise are they partakers of Christ: but rather, to their condemnation, do eat and drink the sign or Sacrament of so great a thing.
Article XXX: Of both kinds
The Cup of the Lord is not to be denied to the Lay-people; for both the parts of the Lord's Sacrament, by Christ's ordinance and commandment, ought to be ministered to all Christian men alike.
Honestly, I don't see how Washington's turning his back on the Lord's Supper can be reconciled with the 39 Articles of Faith. There is sort of an "out" -- that a wicked, unworthy person, is not supposed to take communion. But what does that say about Washington, that for his whole life from revolution to death, he thought himself "[w]icked...such as be void of a lively faith?"
According to these very articles Washington swore to uphold, at least as I read them, a believer, if he doesn't feel "[w]icked...such as be void of a lively faith" is enjoined to take the Lord's Supper.
As Washington's own minister, Dr. Abercrombie put it:
“I cannot consider any man as a real Christian who uniformly disregards an ordinance so solemnly enjoined by the divine Author of our holy religion, and considered as a channel of divine grace.”
Conclusion: Washington's refusal to take communion violated the very oaths he swore to uphold when becoming a Vestryman and Godfather. That is evidence that he didn't believe exactly in "the Doctrine and Discipline of the Church of England, as by law established." If that makes him a hypocrite in the eyes of the pietists, that's a conclusion that they must draw; it's a corner they've painted themselves in.