I agree with what both Brayton and Sandefur have written on Blackstone, religion and the civil law. I'd like to point out how this dispute illustrates how Reason (Athens) and Revelation (Jerusalem) have played a central role in virtually all aspects of Western Society (especially politics), but how ultimately, the public foundation of the United States rests on the sovereignty of Man's Reason and consigns the Truth of Revelation to the realm of private conscience.
The common law is based on the natural law. And natural law is synonymous with what is discoverable by man through Reason. Natural law is an extremely ancient concept. It traces back to Aristotle (so it has Pagan roots) and was adopted into Christendom by Aquinas (and Christian roots). The Anglican Church (I do believe) inherited the natural law from its Catholic roots. And these Christians generally used the natural law to support Church teachings. This pre-Enlightenment natural law existed at a time when Church and State were one, and as Sandefur pointed out, it shouldn't surprise us that religious freedom was not part of the common law of England (or the then understanding of the natural law).
Blackstone understood that the natural law was a type of organic law that trumped all human positive law. It was, in theory, "binding over all the globe, in all countries, and at all times; no human laws are of any validity, if contrary to this; and such of them as are valid derive all their force, and all their authority, mediately or immediately, from this original." This theory was helpful to judges, especially when deciding novel cases with no underlying statute. They could operate under the assumption that they were not "making the law," but rather, "discovering" through the use of their Reason what was already there.
But Blackstone then discusses how man's Reason was corrupt and Revelation could be used to guide reason in discovering what the natural law held. Hence Roy Moore could write (in the case Brayton discusses): "Natural law is the law of nature and of nature's God as understood by men through reason, but aided by direct revelation found in the Holy Scriptures". And it was on this basis then that Christianity could be incorporated into the common law.
This represents more of an "old-school" (Thomist) pre-Lockean, pre-Jeffersonian-Madisonian understanding of the natural law where the concept was used by ecclesiastical Authorities to support Church doctrine and power. But America was founded on a more modern understanding of nature and reason. And by "modern" I mean that "reason" had "discovered" certain errors in the older version of the "natural law" and hence corrected them, or otherwise incorporated "new" discoveries. The "old" (Thomist) version of the natural law justified such things as slavery and burning heretics at the stake and knew nothing of the concept that men existed in a "state of nature" and possessed unalienable natural rights.
Our Founders, the American Whigs that they were, were keen on these "new discoveries" of man's reason which had the effect of pulling the rug from underneath the divine rule of kings and ecclesiastical authorities. As Allan Bloom wrote in The Closing of the American Mind:
From his reflection on the state of nature, Locke drew the formula of Enlightenment, with its particular combination of natural and political science. Its starting point is the untrammeled use of reason....Freedom for man consists in ordering his life according to what he can see for himself through his most distinctive faculty, liberated from the force of tyrants and the authority of lies, i.e., myths. Through unaided reason, man as man, as opposed to the man of this place or time, nation or religion, can know the cause of things, can know nature for himself. p. 163-4.
It had to be, for in order to have rulers who are reasonable, many of the old rulers had to be replaced, in particular those whose authority rested upon revelation. The priests were the enemies, for they rejected the claim of reason and based politics and morals on sacred text and ecclesiastical authorities. p. 257-8
As Sandefur and Brayton pointed out, the writings of the American Founders are replete with natural law references which contradict Blackstone's views on religion and government. Religion, to our Founders, was an unalienable natural right -- a right that was equally possessed, in Jefferson's words, by "the Jew and the Gentile, the Christian and Mahometan, the Hindoo, and Infidel of every denomination," and that men had the unalienable right to worship not just the God of the Bible as he saw fit, but also, again in Jefferson's words, "twenty gods, or no god." Because every man's religion, no matter how unorthodox, is an unalienable right which government must respect, religion is wholly within the realm of man's private conscience or opinion. It logically follows then that Madison and Jefferson would call for a "separation of Church and State" to ensure that such rights are not violated. And the notion that Christianity is part of the common law (hence something judges could enforce) completely contradicts these new natural law discoveries. Hence Jefferson's assertion "that Christianity neither is, nor ever was a part of the common law."
Such a notion was an "error" that "reason" had discovered and corrected, just as Newton's science replaced earlier errors and how Einstein's discoveries replaced some of the things Newton believed.
Now, one could assert that nonetheless, the Lockean-Jeffersonian-Madisonian "updated" version of the natural law is still largely consistent with and complementary towards Biblical Revelation -- that such an understanding, for instance, improved orthodox Christians' ability to "understand" the religious toleration that exists within the pages of the Bible (this is the position that West Coast Straussians, Michael Novak, and some orthodox Protestants would endorse). The East Coast Straussian would argue that, no, Reason and Revelation are not quite so compatible. And many of the "discoveries" of Locke, Jefferson, et al. entirely contradicted the traditional classical/Christian understanding of nature. Leo Strauss held that Locke and Hobbes's "State of Nature" theory was wholly alien to the Bible. And indeed, the notion that men have an unalienable right to worship no God or Twenty God means that men have an unalienable right to break the First Commandment!
Just how compatible the "natural law/natural rights" theory of the Declaration of Independence is with traditional orthodox Christianity, cannot be resolved in this post as it is a subject of great controversy. It is interesting to note that the "natural law/natural rights" theory which founds our government does have a theology that goes with it (see this article by Thomas West). We might even call this a "public theology" or a "civic religion." It should be stressed that this public theology is not orthodox Christianity, rather, it is a natural (meaning discoverable by reason) theology. It holds that there is a Creator, Nature's God, that He grants men unalienable rights, that He created men free and equal, and may intervene when we don't respect each other's natural rights. It could be argued that this theology is, in some ways, consistent with traditional Christianity, but it is not orthodox Christianity. When Jefferson, Adams, Franklin and others explicated the finer details of their theology, it completely contracted orthodox Christianity. Reason discovered that God was unitarian not Trinitarian in nature, that He doesn't send men to Hell for eternity, that good works get one into Heaven, that Pagan religions contain the same Truth as Christianity and are also a valid way to God, that the Bible contains errors and amendments, and that God did not intervene in man's affairs in ways inconsistent with the laws of nature and science.
I point this out because there is an attempt by "Christian Nation" folks to connect our public principles in our public founding documents to orthodox Christianity, as though they "own" it, and, at best, make others who don't operate in such a tradition seem as "outsiders" or worse, argue that they ought not have equal constitutional rights. Justice Scalia endorsed the former view, and argued that while the Abrahamic religions (Judaism, Christianity, and Islam), have full and equal constitutional rights, "[w]ith respect to public acknowledgment of religious belief, it is entirely clear from our Nation's historical practices that the Establishment Clause permits this disregard of polytheists and believers in unconcerned deities, just as it permits the disregard of devout atheists."
Scalia especially focused on the religious views of the first four Presidents as well as founding documents which invoked God. But those Founders and those documents speak in the language of "natural theology." And such a theology, as we have seen, is not consistent with orthodox Christianity in its finer details. Yet, orthodox Christians were not offended by our Founders supplications to God. But the Founders also never contradicted their unorthodox theology when they made such supplications. (This is why George Washington, though he spoke of a warm Providence, almost never publicly mentioned the words "Jesus Christ." It was as if that name was taboo to Washington's lips). So these Founders, when they invoked God, drew a Lowest Common Denominator between their unorthodox natural theology and the orthodox Christianity to which many common folks and ecclesiastical authorities in positions of social power adhered. This was our "civic religion." But it was a very low LCD: There is a God; He grants us unalienable rights; He is concerned about human beings and will intervene, especially if we don't respect the unalienable rights of others and nothing more. None of the above four Presidents ever more specifically defined the attributes of God when publicly acknowledging Him.
In other words, it is not Revealed religion, which is now consigned to the private sphere of society (the realm of "individual conscience" or "opinion") that founds our public order. It is the natural theology, but only invoked in a very vague and simple sense, otherwise, the consciences of orthodox Trinitarians would be "disregarded."