I just found online an important sermon given in 1775 by Samuel Langdon, former President of Harvard University, entitled The Co-Incidence of Natural With Revealed Religion. It's important because it sheds light on the natural religion that forms the basis of American political theology. The Declaration of Independence invokes "the Laws of Nature and of Nature's God" and various folks argue over and misunderstand what this phrase means. One reading holds that the Founders were almost all Deists and this is a stock Deist phrase. And the other holds that the phrase in question is shorthand for the God of the Bible and revelation. Both readings are wrong.
The phrase chiefly refers to reason and sets reason not revelation as the foundation for the United States' political order. The word "nature" as used in this context refers to what is discoverable by reason unaided by scripture. Some have intimated that "the Laws of Nature" refers to reason and "the Laws of...Nature's God" refers to revelation (see for instance, John Eastman) and others have noted if the phrase twice speaks of reason it is redundant. Well the phrase is somewhat redundant because the term nature is used twice: "the Laws of Nature  and of Nature's  God...." [My emphasis.] More likely "Nature's God" was invoked after "the law of Nature" to make the natural law binding. You don't need a God to discover certain immutable principles through nature. E = MC2 is part of the "laws of nature" regardless whether God exists. But that just tells us what "is" not what "ought to be." Invoking God, in this sense helps close the is/ought gap. The concept of the laws of Nature's God is more properly termed "natural religion" (as opposed to "natural law") and that is what man can know about God and His universe from reason unaided by scripture.
And this is where Langdon's sermon proves useful. As he explains natural religion:
The religion of nature, considered in the most perfect view, is that which we suppose investigable by the natural powers of the human mind without the assistance of any revelation from heaven.
Langdon then goes on to explain "[w]hat system of religion reason alone would trace out."
Langdon was as far as I know an orthodox Trinitarian Christian (but also one of the patriotic Whig preachers). This illustrates that it wasn't just Deists but orthodox Christians as well (and those folks like America's key Founders who were in between Deism and orthodox Christianity) who believed in natural religion. So the appeal to natural religion was not a Deistic attempt to exclude orthodox Trinitarians from the Declaration of Independence but an attempt to unite Deists, Unitarians, Trinitarian Christians, and potentially all other theistic world religions. As the Masonic Book of Constitutions put it, it was "that Religion in which all Men agree."
Indeed Freemasonry united Deists, theists and orthodox Christians during the Founding era. Theists George Washington and Ben Franklin were Freemasons. Theist Thomas Jefferson and Deist Thomas Paine, though I don't believe they ever joined nonetheless spoke highly of the craft. And lots of orthodox Trinitarian Christians likewise were members. The very orthodox Presbyterian minister Samuel Miller was a Freemason and addressed its relationship with Christianity. Peter Lillback, in his study on Washington's religion, reproduces part of one of Miller's sermons:
Masonry, as such, and according to its original plan, appears to be founded on natural religion. Hence the institution is found among all nations, who believe in one God, and the accountableness of man to him, as a moral Agent, and an immortal being.
-- quoted from George Washington's Sacred Fire, p. 505.
Natural religion served as a politically useful mechanism to unite anyone but an atheist (who were virtually non-existent at the time of the US Founding) under the precepts of what all rational men can agree on. The US Founders needed a way to take the sectarian squabbles out of politics while still remaining "religious" and natural religion (not the revealed religion of orthodox Christianity) served that purpose.
Orthodox Christians like Samuels Miller and Langdon could feel comfortable with natural religion because, as they noted, reason and revelation (natural and revealed religion) agreed.
Yet, the issue of natural religion and the US Founding poses a few challenges for a "Christian America" reading of history. First, even though many Christians of that era purported compatibility between natural and revealed religion, America's Founders when declaring independence invoked natural religion only, not revealed religion. Hence "reason" not "Christianity" forms the bedrock component of the American Founding's political system.
Indeed, because the Bible says nothing about men being endowed with unalienable rights and government's purpose being to secure them, it was more honest for America's Founders to "channel" such ideas through philosophical naturalism or theological rationalism. The Declaration of Independence truly was the product of man's reason, not biblical ideas.
For Christians, natural religion also risks importing non-biblical ideas (again like those found in the Declaration of Independence) and holding them up on the same level as sacred scripture. Indeed orthodox Christians like Samuel Langdon, Ezra Stiles and many others, in their political sermons, read Enlightenment naturalistic ideas into the Bible's text when they erroneously declared such things as the Ancient Israelites had a republic and the law of Moses was voluntarily adopted by the Jews. In short, one can argue, whatever the compatibility, natural religion is not good for the purity of Christianity's orthodoxy, exactly what Francis Schaeffer argues here, where he rails against Aristotle's influence on Christianity. The problem for Schaeffer is that it makes his Christianity anathema to the American Founding, which promoted Aristotelian, naturalistic theology often under the auspices of "rational Christianity."
Finally, natural religion, like Freemasonry (at its mildest) is problematic for orthodox Christians because it promotes, as this article on Freemasonry notes, indifferentism.
Indifferentism is the heretical belief that all religions are equally legitimate attempts to explain the truth about God which, but for the truth of His existence, are unexplainable. Such a view makes all truths relative and holds that God can be equally pleased with truth and error. Because Christians believe that God has definitively revealed Himself in the person of Jesus Christ, and desires that all men come to the knowledge of this truth, indifferentism is incompatible with Christian faith. Jesus said, "I am the way, the truth, and the life; no one comes to the Father, but by me." (John 14:6).
Freemasonry's teachings and practices also result in syncretism which is the blending of different religious beliefs into a unified whole. This is evidenced most especially by Masonry's religious rituals which gather men of all faiths around a common altar, and place all religious writings along side the Bible on the Masonic altar. This is also demonstrated by the Lodge's prayers and its unique names and symbols for God and heaven. Syncretism is the logical consequence of indifferentism.
America's key Founders [Washington, Adams, Jefferson, Madison, Franklin, et al.], I've long argued, held precisely to this "indifferentism" or syncretism in religious matters such that they ceased to be "Christian" in an orthodox Trinitarian sense. They were also disproportionately Masons. One could still believe in these non-Christian, Enlightenment ideas (Freemasonry, natural religion, philosophical rationalism) and still be an orthodox Christian. Yet, it's like playing with fire. Overindulging in these ideas could lead folks raised in traditional Christian homes away from the faith, which is, I believe, exactly what happened with America's key Founders.