I think it’s commonly understood that the ideals framing the Declaration of Independence were the result of the Enlightenment Age and Protestant notions, which were somewhat at odds with Catholic political thought. However, there’s an interesting lecture at http://www.udallasnews.com/media/paper743/news/2005/10/12/News/Guest.Lecturer.Reminds.Of.Americas.Catholic.Roots-1016918.shtml by Scott McDermott of Vanderbilt University who points to John Carroll, one of the signers of the Declaration, as an introducer of Catholic thought to the new colonial government.
McDermott explained that before the Reformation, common law was understood to be based on natural law. St. Thomas taught that if a law 'conflicts with the law of nature it will no longer be law but rather a perversion of law.' He and others also held that when a ruler becomes a tyrant and forces his citizens to act contrary to natural law he abdicates his sovereignty and sovereignty reverts to the people who must then delegate a new sovereign.
The lecturer says that such Catholic ideals were "unwittingly" brought to the Colonial mindset. He goes on to say,
"The culmination of the American rediscovery of natural law and natural rights is, of course, Jefferson's Declaration of Independence, with its appeal to the "Laws of Nature and of Nature's God," and its listing of the self-evident natural rights to justify resistance to tyranny," McDermott said. "Anglo-American republican liberalism embraced the natural law, not the general will, as its liberating principle."
I think this is interesting and points to something rather telling in modern times. Democracy without regard for such "natural rights" is not what the Founders envisioned, and yet seems to be precisely what the United States is leaning towards.
I have heard about McDermott and he has a point. His arguments are clearly borrowing from J.G.A. Pocock’s The Machiavellian Moment where he argues that the Florentine Republic (Catholic of course) served as a political model for Jefferson and Madison. However, the ideas that formulated the synthesis which created the Declaration of Independence and later the Constitution, came from the Scotish Enlightenment (Berkeley) and of course John Locke (though many historians wish to efface him from the formula all together). Other influences were Lord Bolingbroke who received quite a large amount of affection by Jefferson and I have to toss in Montesquieu. Rousseau should not be forgotten entirely either. Locke covered the arguments of natural law quite well saying that moral law is paramount to the vitality of virtue and necessary for government to be efficacious. Natural law would fit more into Machiavelli’s world and the world of Thomas Hobbs. Of course Aquinas would have made a good Calvinist since he found Predestination and Providence to be complimentary aspects of the human condition. The synthesis could also include David Hume where human inquiry is involved, but this is much harder to prove. I think Calvinism and evangelicalism played a crucial role by the time of the Constitutional Convention, but perhaps not so much for the Declaration of Independence. For constitutional arguments that led to the Declaration of Independence, see John Phillip Reid. He has a book on the Constitutional Arguments of the Revolution which basically proved that the colonists felt their charter remained in the hands of the king and that Parliament held no claim to them. Also, Reid argued that the colonists were still clinging to property rights instead of title rights as Parliament was so doing. In other words, the colonists viewed their rights rested in material possession (and so with the king who owned all of the land), whereas Pariament held the title of the sovereign via the Commons which "virtually" represented colonists 3000 miles away. Good stuff really....check him out.
From what you’re saying – and as I would imagine – there were many philosophical influences that played a part in the Founding documents. Some of these are probably more obvious than others. Lockean thoughts are clearly involved (right of people to revolt); perhaps Rosseau with his social compact theory; Hobbes with his views on man coming together to have a common judge, etc. That said, I think there were certain ideals of those thinkers that are actually contrary to the Founding ideals. For example, Rousseau and Hobbes both believed that no law could be unjust. This seems to be contrary to the Aristotelian and Thomistic view of “natural justice”, which I think entails the view of “unalienable rights”.