Robert P. Kraynak's Christian Faith and Modern Democracy is another book (there are many) written by a conservative Christian (he's a Catholic) that debunks the "Christian Nation" thesis.
Dr. Gregg Frazer heavily relies on this work in his Ph.D. thesis. Kraynak's work is important in illustrating the context which neither the Christian right nor the secular left well understand. Though the Bible was not cited in our Founding documents (Declaration, Constitution, and Federalist Papers), many orthodox Christians existed in the population and the American Revolution was a common subject in the pulpits. Indeed pamphlets reproducing such sermons abounded back then. Yet, much about our Revolution could seem to conflict with the Bible and the traditional understanding of Christianity. Most notably, Romans 13 where Paul tells believers in no uncertain terms to obey the civil rulers. And not just "Godly" rulers -- the ruler to whom Paul refers was the Pagan psychopath Nero. Thus, propagandistic arguments had to be made arguing that the principles of the Revolution really were consistent with the Bible and Christianity. Hence, many citations to the Bible and Christianity arguing their compatibility with the principles of the Revolution, the Constitution, and Republicanism. But in making this case, the history of Israel especially had to be radically rewritten. And this is why the "Christian Nation" crowd can offer quotations, taken out of context, which seem to show that the Founders based their claims on liberty on the Bible.
For a recent example, Don Feder writes:
Our form of government is based on the Bible. At the dedication of the Bunker Hill Monument (1843), Daniel Webster declared that the Bible "is also a book which teaches man his own individual responsibility, and his equality with his fellow-man."
The Bible contains the seeds of our current conception of equality under the law and human rights. (The American Revolution was preached from colonial pulpits. The anti-slavery movement started in the churches of New England.) That's why the Western world pioneered the abolition of slavery. That's why the Islamic world still has it.
And the Christian Nation crowd loves to point out how the Liberty Bell quotes Leviticus.
But they fail to understand a) that the people often making the claims were not Christians and rejected much written in the Bible and contained in orthodox Christianity, and b) that the Bible in general and those quotations in particular, understood in context, arguably don't specifically support the Declaration, the Constitution, or founding republican ideals. In short, one could argue that the patriotic preachers and republican Whigs actually abused the Bible in service of their cause, because hey, desperate times called for desperate measures.
Kraynak's book is one of the few places that well-understands this historical context. The following quotes from Dr. Frazer's thesis quoting Kraynak's book:
First, as Kraynak pointed out, "the biblical covenant is undemocratic: God is not bound by the covenant and keeps His promises solely out of His own divine self-limitation." Second, "(t)he element of voluntary consent is missing from the covenant with Israel....There is nothing voluntary or consensual about the biblical covenant; and the most severe punishments are threatened by God for disobedience." Third, "insofar as the covenant with Israel sanctions specific forms of government, the main ones are illiberal and undemocratic;" including patriarchy, theocracy, and kingships established by divine right. Fourth, "the Bible shows that God delivers the people from slavery in Egypt and supports national liberation, not for the purpose of enjoying their political and economic rights, but for the purpose of putting on the yoke of the law in the polity of Moses." Fifth, "the content of the divine law revealed to Moses consists, in the first place, of the Ten Commandments rather than the Ten Bill of Rights, commanding duties to God, family, and neighbors rather than establishing protections for personal freedom." Finally, the combination of judicial, civil, ceremonial, and dietary laws imposed on the people "regulate all aspects of religious, personal, and social life." The history of Israel, therefore, had to be radically rewritten to provide support for the demands of political liberty and for republican self-government.
-- Kraynak, 46-49 quoted in Frazer, "The Political Theology of the American Founding," Ph.D. dissertation, 18-19.
So when, for instance, John Adams stated about the Bible, "It is the most Republican Book in the World, and therefore I will still revere it," keep what you have just read in mind as well as how Adams' theology -- his unitarianism, universalism, and rejection of Biblical inerrancy -- otherwise conflicted with orthodox Christianity.
The colonies did indeed invoke the Biblical Covenant when founded in an earlier era. But since that concept didn't work for the "Novus Ordo Seclorum," such was replaced by the Lockean social contract. This isn't to say that covenant theology and colonial charters had nothing to do with our Founding from 1776-1789. Those colonial charters were to some extent, experiments with self government and did anticipate some of the ideas of our national founding. Even Bernard Bailyn recognizes that Biblical principles/Covenant theology were one of the sources from which our Founders drew when positing the principles of the Declaration/Constitution. However, as men of the Enlightenment, our Founders approached the colonial charters and the Bible in a cafeteria manner, picking and choosing what they thought "rational" and discarding the rest. So when it came time to pick and choose from "covenant theology," they left out the very heart -- the covenants themselves (neither the Constitution, nor the Declaration or Federalist Papers contain or call for the Biblical Covenant!) -- and replaced them the with the social contract and Art. VI of the Constitution's "no religious test" clause.