That's one of the themes in Gary North's "Conspiracy In Philadelphia." I am going to reproduce this passage from Dr. North's E-Book because its analysis is so spot on:
We do not find authoritative references to the Bible or church history in either The Federalist Papers or the Antifederalist tracts. Adrienne Koch’s compilation of primary source documents, The American Enlightenment, is not mythological, even though it is self-consciously selective.36 There was an American Enlightenment, though subdued in its hostility to Christianity.37 Jefferson, after all, kept hidden his cut-up, re-pasted New Testament, purged of the miraculous and supernatural; he knew what his constituents would have thought of such a theology.38 He refused to publish this book, he told his friend, Christian physician Benjamin Rush, because he was “averse to the communication of my religious tenets to the public, because it would countenance the presumption of those who have endeavored to draw them before that tribunal, and to seduce public opinion to erect itself into that inquest over the rights of conscience, which the laws have so justly proscribed.”39 That is, if word got out to the American voters, who were overwhelmingly Christian in their views, regarding what he really believed about religion, he and his party might lose the next election, despite a generation of systematic planning by him and his deistic Virginia associates to get Christianity removed from the political arena in both Virginia and in national elections. (The book was not made public until 1902. In 1904, the 57th Congress reprinted 9,000 copies, 3,000 for use by Senators and 6,000 for the House.40 It was a very different America in 1904.)
The Framers rhetorically appealed back to Roman law and classical political models in their defense of the Constitution. Madison, Jay, and Hamilton used the Roman name “Publius” in signing the Federalist Papers. Publius was a prominent founder of the Roman Republic. The Antifederalists responded with pseudo-Roman names. Yet both groups were heavily dependent on late seventeenth-century political philosophy, as well as on early eighteenth-century Whig republicanism....41 They shared a common universe of political discourse, and trinitarian Christianity was what both sides were attempting to downplay. The political discourse of the age was dominated by classical allusions, not by Hebraic ones....
....What we must understand is that the U.S. Constitution is in large part a product of a rhetorical Enlightenment appeal back to the Greco-Roman world, yet it was in fact something quite modern: specifically, a reaction against the Puritanism of both seventeenth-century American colonialism and the Puritanism of the Cromwellian revolution of 1642–60.
To what extent was this verbal appeal back to Rome rhetorical? [Thomas] Pangle believes, as I do, that the Framers were essentially “moderns” rather than “ancients.” They were far more influenced by late seventeenth-century social thought than by the events of Roman history, let alone classical political philosophy, which had little impact on them except in a negative sense. “Generally speaking, the ancients, in contrast to the American Founders, appear to place considerably less emphasis on protecting individuals and their ‘rights’ – rights to private property and family safety, to property, to freedom of religion, and to the ‘pursuit of happiness.’”45 Also, he argues – I believe correctly – that the classical philosophers put virtue above fraternity and liberty.46 The Framers, while they discussed the need for virtue and religion – always carefully undefined – did so as defenders of political and economic freedom. Virtue was therefore instrumental for them – a means of achieving social stability and progress, liberty and security.47
This was also their view of religion. In this, they were not fundamentally different in principle from Robespierre, who established a formal civic religion of nature and reason in the midst of the Terror in 1794. De-Christianization was morally debilitating, Robespierre concluded; it had to be followed by the establishment of a new civic religion.48 He knew that men needed to believe in God’s sanctions in order to keep them obedient. Talmon calls this impulse “cosmic pragmatism.”49 The major figures among the Framers were wiser men than Robespierre, and more influenced by traditional Christianity, but they were Enlightenment men to the core. Their veneer and their constituencies were different from those of the French Revolutionaries, but not their theology. Their religion was civic religion. The difference is, they saw civic religion as a decentralized, individual matter rather than as a state affair; it was to aid the national government but not be part of the national government. John Adams, a theological unitarian, wrote in his autobiography, presumably for himself and not the electorate:
"One great advantage of the Christian religion is that it brings the great principle of the law of nature and nations, Love your neighbour as yourself, and do to others as you would that others should do to you, to the knowledge, belief and veneration of the whole people. Children, servants, women and men are all professors in the science of public as well as private morality. No other institution for education, no kind of political discipline, could diffuse this kind of necessary information, so universally among all ranks and descriptions of citizens. The duties and rights of the man and the citizen are thus taught, from early infancy to every creature. The sanctions of a future life are thus added to the observance of civil and political as well as domestic and private duties. Prudence, justice, temperance and fortitude, are thus taught to be the means and conditions of future as well as present happiness."50
Not a word about the atonement; not a word about the sacraments: the entire passage is geared to the requirements for public morality. The churches are viewed as effective educational institutions; no other institution could accomplish this task more effectively. Hence, Christianity is a good thing socially. The whole perspective is civic. pp. 27-32.