At Positive Liberty -- one of my other websites -- Tom Van Dyke left a thoughtful comment on whether the God of the American Founding is the God of Abraham:
I’m not aware of any other monotheistic, providential Creator God that can be remotely construed to endow man with certain unalienable rights.
All this talk of syncretism must acknowledge that the syncresis took place within a very narrow milieu of a Judeo-Christian European culture with, admittedly, the acknowledged philosophical influence of the Enlightenment and the Greeks [with a dash of the Romans thrown in].
But there is no new God of the Enlightenment except perhaps for man himself, and the gods of the Romans and Greeks are nowhere to be found here except on the edges, and only rhetorically.
The God of the Founding is not a new one, fabricated from whole cloth. He may not be Abraham’s, strictly speaking, but He is none other, either.
I see his point -- that though America's Founders pretended the Native Americans with their "Great Spirit," the Hindus, the pagan-Greco-Romans, and other non-Judeo-Christian faiths worshipped a common "Providence," carefully examining the attributes of these non-Judeo-Christian deities belies such a notion. Michael Novak makes a similar point that an active, personal, intervening monotheistic God is uniquely characteristic of Judaism and Christianity (he doesn't add Islam, but I will).
Still, in carefully examining the attributes of America's Founders' benevolent unitarian deity, one must ask is this the authentic biblical God? I've noted before that since America's key Founders believed man's reason superseded biblical revelation, this was the God of the Bible, minus everything written in the Bible that America's Founders found irrational like God's wrath, judgment, and jealously (plus other things added in like the fact that men have unalienable rights to life, liberty, equality, property, the pursuit of happiness and to revolt against civil governments that don't secure such unalienable rights, ideas for the most part, not found in the Bible).
Dr. Gregg Frazer, a traditional evangelical, makes the case that America's Founders and the philosophers and theologians they followed remade the biblical God over in their image to be more man centered. This was not modern day secular humanism, which is atheistic or positivist in its outlook; but classical era, theistic humanism where God becomes more man centered (as opposed to man being God centered).
Accordingly, figures like Jonathan Edwards and George Whitefield represented the "old way" which was the authentic and traditionally Christian way of man being God centered. But America's Founders followed men like John Locke, Samuel Clarke, Joseph Priestley, Jonathan Mayhew, Charles Chauncy, and others who were Christian heretics, theists, and rationalists who posited theistic humanism. Even many of the orthodox Christians who influenced the American Founding like John Witherspoon, Samuel Langdon, and Ezra Stiles, when they preached politics, did not speak in authentically Christian terms, but rather borrowed from the theistic rationalists.
This approach of course, concedes that the traditional conservative Christianity that stresses such notions as the Trinity, eternal damnation, and Christ as the only way, is indeed authentic Christianity! One could also make the case that the Founders' Christianity which denied the Trinity and eternal damnation, is true, authentic Christianity. If one concedes that you can disbelieve in the Trinity and believe in universal salvation and still be a Christian, I'd have a hard time arguing that Washington, Adams, Jefferson, Madison, and Franklin were not Christians. Personally, I would concede that these key Founders qualify as "Christian heretics." They presented their ideas under the auspices of Christianity, not Deism.
In his PhD thesis Dr. Frazer quotes some telling passages from earlier academics on this change in perspective and Charles Chauncy, one of the most important Founding-era, pro-revolutionary ministers, a unitarian and a universalist, and the quintessential theologian who remade God to be more man centered. As he quotes one such scholar of religion:
Before the good of man consisted ultimately in glorifying God; now the glory of God consists in the good of man. Before man lived to worship and serve God, and now God lives to serve human happiness.
-- Joseph Haroutunian, "Piety versus Moralism: The Passing of the New England Theology," 145.
Frazer also points to a book by James W. Jones entitled "The Shattered Synthesis" where Jones compares the theology of Jonathan Edwards to that of Charles Chauncy's:
For Edwards, God's actions must be consistent with God's own nature and intentions; for Chauncy, God's actions must be consistent with what he calls "the common happiness." For Edwards, God's actions must be consistent only with his own glory. For Chauncy, since God's benevolence is directed not toward God himself but primarily toward creation, God's actions must be consistent with the good of creation.
-- pp. 168-69, quoted in Frazer, "The Political Theology of the American Founding," p. 305.
As a conservative evangelical, Frazer terms this "a brand of humanism in which even God is man centered." Ibid. A theistic humanism as part of theistic rationalism, the true political theology of America's Founding, not orthodox Christianity.
The thrust of Frazer's thesis, with which I strongly agree, is that when America's Founders invoked God for their republican principles, they didn't invoke the God of Edwards and Whitefield, but of Chauncy, Mayhew, Priestley et al. And this "republican" God was not the authentic God of the Bible or Christianity (for that, you'd have to go to Edwards and Whitefield). However, I do note that Chauncy, Mayhew, Priestley, and most of America's key Founders did consider themselves Christians. And that even today some theologically liberal or moderate Christians who don't believe the Trinity, eternal damnation, and infallibility of the Bible are prerequisites to the true Christian faith could argue (and would argue if they understood what America's key Founders really believed) that theistic rationalism is just a form of theologically liberal Christianity, what traditional orthodox Christians consider "heresy."