Unfortunately I can't give a clear cut answer as to exactly what GW thought of the Bible, because he never told. However, I can make some general observations.
First, the record does not demonstrate that GW thought the Bible the inerrant infallible Word of God. It's possible; but I doubt it. It's even MORE unlikely that Washington, like the strict Deists, didn't believe in ANY of the Bible as divine revelation.
Washington quoted from the Bible quite a bit. Indeed in his new book, The Political Philosophy of George Washington, Jeffry H. Morris claims that GW quoted from the Bible more than any other source, including Joseph Addison's Cato, which came in second.
I'm enjoying reading Morrison's book and I agree with the overwhelming majority of what I have so far seen. He argues GW's political philosophy was a synthesis of classical Greco-Roman ideas, moderate British Enlightenment ideas, and Protestant Christianity. After Bernard Bailyn, I noted something similar in my entry on GW for the Cato Institutes's Encyclopedia of Libertarianism. However Bailyn/I analyze the synthesis with 5 prongs instead of 3. In the entry I wrote:
Harvard professor and historian Bernard Bailyn traces the ideological origins of America's founding-era republican thought to the following principal sources: Classical Greco-Roman antiquity; Biblical theology; English common law; Enlightenment rationalism, and the writings of British Whig theorists like Algernon Sidney, John Locke, and Joseph Priestly.
And George Washington's personal political philosophy was the same as the American Founding's (or vice versa). Moreover, some of these categories bleed into one another.
I'm still looking in the footnotes to Dr. Morrison's book to see how he statistically concluded GW quoted the Bible more than any other source (like Cato). Though, to respond, 1) if the other classical sources (Aristotle, Cicero, Seneca) were added to Cato, would GW's classical quotes exceed his biblical quotes? And 2) simply quoting the Bible, even a lot, doesn't demonstrate one's opinion about the Bible's authoritative truth. Just as Shakespeare dramatically changed English language communication, the Bible changed Western Civilization's communication. Even today, atheists regularly quote the Bible to illustrate a point, as did strict Deists like Thomas Paine during the Founding era. Ben Franklin so mastered biblical style that when he transitioned from quoting the Bible to improvising his own words while pretending to quote the Bible, most of his everyday hearers couldn't tell the difference. Franklin used to do that as a parlor trick, by the by.
And Washington rarely if ever quoted the Bible in a clear authoritative way. But I am going to reproduce some of Washington's closest to authoritative dealings with scripture. I first want to deal with an address I've seen advanced as evidence of Washington's traditional Christian piety. In an undelivered First Inaugural Address, Washington supposedly said:
The blessed Religion revealed in the word of God will remain an eternal and awful monument to prove that the best Institutions may be abused by human depravity; and that they may even, in some instances be made subservient to the vilest of purposes.
The problems with this quote are 1) it probably wasn't written by Washington, but by Colonel Humphreys; 2) Washington didn't give this as an address but canned it; and 3) he did so probably because it was not characteristic of his thoughts, so much so that Jared Sparks, one of the first biographer/archivists of Washington's, tore up the address into pieces and gave those pieces to his friends; that's why the address exists in fragment form. The following is what Paul F. Boller writes in "Not so!: popular myths about America from Columbus to Clinton":
The Humphrey's-Washington draft [of Washington's first inaugural speech] exists only in part today. What happened to it chills the blood of present-day historians and biographers. In 1827, years after Jared Sparks, Unitarian minister and editor of the North American Review, who was preparing to publish a collection of Washington's writings, came across the speech in Washington's handwriting and decided to suppress it, partly because he thought some passages in it were a bit strange and partly because he knew it had eventually been discarded. On May 22, Sparks asked the aging James Madison, who had thirty-eight years before, told him: "I concur without hesitation in your remarks on the speech of seventy-three pages and the expediency of not including it among the papers selected for the press. Nothing but extreme delicacy towards the author of the draft, who was no doubt Colonel Humphreys, can account for the respect shown to so strange a production." Sparks then blithely cut the manuscript into pieces and handed out parts of it to autograph collectors anxious to own something in Washington's own handwriting. Years later scholars attempted to reassemble the fragment, but succeeded in recovering only a third to one-half of the original. The anfractuosities of what survives are probably Sparks's doing, not Washington or Humphrey's.
What was a more mainstream and characteristic utterance of Washington on revelation (again not written by him, rather by David Cobb, but given under his name) is is 1783 Circular to the States:
The foundation of our Empire was not laid in the gloomy age of Ignorance and Superstition, but at an Epocha when the rights of mankind were better understood and more clearly defined, than at any former period, the researches of the human mind, after social happiness, have been carried to a great extent, the Treasures of knowledge, acquired by the labours of Philosophers, Sages and Legislatures, through a long succession of years, are laid open for our use, and their collected wisdom may be happily applied in the Establishment of our forms of Government; the free cultivation of Letters, the unbounded extension of Commerce, the progressive refinement of Manners, the growing liberality of sentiment, and above all, the pure and benign light of Revelation, have had a meliorating influence on mankind and increased the blessings of Society.
Now, even though Washington, in his other public and private sentiments, rarely if ever discussed what he thought of "revelation," I think we can stretch and conclude this was mainstream American Founding/Washington thought. Indeed, I quoted this in my entry on Washington for the Cato Institutes's Encyclopedia of Libertarianism. However I don't see this passage as proving Washington thought the Bible infallible or inerrant, but rather inspired in *some* sense. I certainly think this passage is consistent with the idea of the Bible as partially inspired. For one, the terms "pure and benign light" are enlightenment qualifers and perhaps indicate a deistic understanding of the Bible/the Christian religion.
As James Hutson commented on James Madison, in 1833, terming the Christian religion the "best & purest religion":
This...sounds very much like the deistical maxim, frequently indulged by Jefferson, that the "pure" religion of Jesus had been unconscionably corrupted by the apostle Paul and the early church fathers.
In preparing notes for the Memorial and Remonstrance, Madison discussed these two different approaches to the Bible/Christianity, the more "orthodox" approach that deemed the entire Bible inspired, and the "other" (more "deistic?", "unitarian?") approach that held "parts" of the Bible inspired:
5. What books canonical, what apocryphal?...
6. In what light are they to be viewed, as dictated every letter by inspiration, or the essential parts only ? Or the matter in general not the words?
In other words, contra the idea that you either accepted all of the Bible or none of it, there was a middle ground position in "Christendom" during the Founding, adhered to by many elite Whigs (many of whom unitarians) that believed "essential parts" only to be divinely inspired, or the Bible in a general sense, not each and every word, as inspired. Washington's utterances in the Circular are compatible with both the orthodox and the more deistic-unitarian understanding of scripture.
Washington's personal letters likewise give no definitive answer about his views of scripture. He oft-quoted the Bible along with classical and enlightenment sources, but did not disclose which sources he held the most authoritative. The following, to MARQUIS DE CHASTELLUX, April 25[-May 1], 1788, likewise illustrates the way GW approached these different ideological sources in his private letters:
While you have been making love, under the banner of Hymen, the great Personages in the North have been making war, under the inspiration, or rather under the infatuation of Mars. Now, for my part, I humbly conceive, you have had much the best and wisest of the bargain. For certainly it is more consonant to all the principles of reason and religion (natural and revealed) to replenish the earth with inhabitants, rather than to depopulate it by killing those already in existence, besides it is time for the age of Knight-Errantry and mad-heroism to be at an end.
That passage is instructive of Washington's personal and political philosophy, and quite amusing. It deals not only with sexual innuendo [making love, under the banner of Hymen], but also quotes Greco-Romanism [the infatuation of Mars], along with Enlightenment [the principles of reason and religion (natural)], and the Bible/Christianity [(and revealed)]. It also does not instruct which of these sources trumps the others.