My co-blogger at American Creation, King of Ireland, is exploring some of the reformed roots of America's Founding. There's no doubt there was a powerful Christian influence, both reformed and natural law (arguably implicitly Catholic).
However there were lots of other things as well. "Christian" is just one of five of Bernard Bailyn's ideological sources of the American Founding, the others being British Common law, Whig, Greco-Roman, and Enlightenment. And keep in mind these sources bled into one other and many of ideas and thinkers behind them, John Locke, for instance, "fit" into more than one category.
What I see as Enlightenment about the American Founding -- why I think it dominated aside from the fact that they were living in the age of Enlightenment -- is that the Founders used their reason to pick and choose from what they wanted (i.e., the Bible, the Greco-Roman tradition, Calvin, Blackstone, traditional natural law teachings) and discard the rest.
With Calvin for instance, whatever it is they might have taken from him, they didn't agree with his teaching that even the worst of tyrants properly command divine submission from subjects.
Now, there are some later Calvinists who seemed more generous than Calvin on resistance theory, Rutherford, Ponet, et al. However even they are little cited by the Founders and ministers and philosophers they followed. I do recall John Adams reciting some of those names in classic Enlightenment fashion, making nominal reference to them as he vomited up names from various ideological sources.
Samuel Rutherford was cited by America's Founders little if at all. But if he did influence them, again, it was in a qualified Enlightenment sense where they took from him what they wanted and rejected what they didn't like. For instance, Rutherford (like most of the other "Calvinists") was horrific on religious liberty issues. Here is Rutherford on Servetus' execution:
It was justice, not cruelty, yea mercy to the Church of God, to take away the life of Servetus, who used such spiritual and diabolic cruelty to many thousand souls, whom he did pervert, and by his book, does yet lead into perdition.
This is anti-American Founding 101.
Likewise with Blackstone, supposedly the origin of the concept of "Nature's God." Ironic in that the Declaration of Independence is an anti-Blackstonian document. Blackstone taught, in no uncertain terms, when the King and Parliament (i.e., British Rule or the particular way in which they split power) take action, no power on Earth can undo it.
Here are Blackstone's exact words on Parliament's power:
It can, in short, do every thing that is not naturally impossible; and therefore some have not scrupled to call it’s power, by a figure rather too bold, the omnipotence of parliament. True it is, that what they do, no authority upon earth can undo.
As Gary North put it: “Blackstone was wrong: beginning eleven years later, the American colonies undid a lot of what Parliament had done.” North also properly observed: "They [America's Founders] took what they liked from his [Blackstone's] system and ignored the rest." [See the following, pgs. 20, 102.]
Likewise with Locke, the most important philosopher who influenced America. He didn't believe in extending religious liberty to atheists or Roman Catholics. But Jefferson explicitly noted, where Locke stopped short, they would go further.
And they did.