Noah Webster is one of the "Christian Nation" crowd's favorite founders to quote, because in the 19th Century, many of his words seem to support their ideal. Yet, as Tom Peters noted on this page, almost all of those quotations come from after 1808 when "Webster underwent a profound religious conversion that changed both his politics and his religious outlook." He "bec[a]me skeptical of democracy, distrustful of government, and far more sympathetic to an alliance between church and state."
Before that time, most importantly when the US Constitution was being framed and ratified, he supported separation of church and state along the lines of Madison's and Jefferson's understanding of the concept.
I've never studied Webster's religious beliefs in detail. I know that in the 1800s when he was saying the things that David Barton et al. like to quote, he was a devout orthodox Christian. Perhaps his political change of mind was precipitated by a bona fide religious change of mind and before that he was, like America's key founders, a theistic rationalist.
Besides the writers of Federalist Papers, a number of notable founders, including Webster, explicated their understand of the meaning of the Constitution's text, and the Liberty Fund reproduces much of these writings. The evidence from these contemporaneous sources shows the Bible/Christian principles were little on the mind of the men who framed the Constitution at the time of the framing. I wouldn't sweep too far and assert the Bible/Christianity had no influence. The founders operated in a culture where churches played an important role and where the Bible/the Christian religion shaped morals, language, and literature for one thousand and several hundred odd years. Simply, I assert the founders did not view themselves as inspired Christians, looking primarily to the Bible to construct a "Christian Nation." This is an historical fiction, not taken very seriously in the academy, but damaging the minds of many in the home schooled crowd.
Harvard Professor Bernard Bailyn in his masterful study of The Ideological Origins of the American Revolution identifies the following principle philosophical sources of America's founding (note, he stresses these as sources of America's Revolution, and though some differences may exist between the thought that produced the Revolution and that which produced the US Constitution, I have concluded that Bailyn's analysis generally applies to America's founding overall, that indeed it perfectly encapsulates American-Whig-republican thought of that era): 1) Biblical/Christian principles; 2) Classical/Greco-Roman principles; 3) English common law principles; 4) Enlightenment rationalist principles; and these 4 seemingly disparate strands of thought were synthesized in a vast body of preexisting literature by 5) British Whig-dissenters like Milton, Sidney and Locke from the earlier era, and Priestly, Price and Burgh who were contemporaries of America's founders. These British Whigs and America's founders were Enlightenment thinkers and as such they viewed Enlightenment rationalism or "man's reason" as the ultimate lens (or ultimate trump) through which all principles were to be viewed.
Now, this thought did have theological implications, and almost all of the thinkers were theists of some sort. However, the theology that best complemented the synthesis was not orthodox Christianity, but theistic rationalism. That's why Gregg Frazer refers to it as The Political Theology of America's Founding (the title to his Ph.D. thesis).
In any event, the following from Noah Webster pontificating on the principles of the US Constitution perfectly illustrates Bailyn's analysis. He refers to the Constitution as establishing an "Empire of Reason." It doesn't get any clearer than that that Webster believed Enlightenment or man's reason provided the ultimate foundation for the new United States of America. Webster views Judeo-Christianity as just another ideological source from which man's reason can pick the "rational" parts, and places Moses along side a pantheon of pagan law givers like Fohi, Confucius, Solon, Lycurgus, Numa, Mango Capac, Zamolxis and Odin. The entire thing is well worth reading. The following is just the introduction:
Of all the memorable eras that have marked the progress of men from the savage state to the refinements of luxury, that which has combined them into society, under a wise system of government, and given form to a nation, has ever been recorded and celebrated as the most important. Legislators have ever been deemed the greatest benefactors of mankind—respected when living, and often deified after their death. Hence the fame of Fohi and Confucius—of Moses, Solon and Lycurgus—of Romulus and Numa—of Alfred, Peter the Great, and Mango Capac; whose names will be celebrated through all ages, for framing and improving constitutions of government, which introduced order into society and secured the benefits of law to millions of the human race.
This western world now beholds an era important beyond conception, and which posterity will number with the age of Czar of Muscovy, and with the promulgation of the Jewish laws at Mount Sinai. The names of those men who have digested a system of constitutions for the American empire, will be enrolled with those of Zamolxis and Odin, and celebrated by posterity with the honors which less enlightened nations have paid to the fabled demi-gods of antiquity.
But the origin of the AMERICAN REPUBLIC is distinguished by peculiar circumstances. Other nations have been driven together by fear and necessity—the governments have generally been the result of a single man’s observations; or the offspring of particular interests. IN the formation of our constitution, the wisdom of all ages is collected—the legislators of antiquity are consulted—as well as the opinions and interests of the millions who are concerned. In short, in it an empire of reason.