It's important to note, whether countering the claims of the likes of Roy Moore that the God of the civil religion, that is the God to whom the Founders made their public supplications, was a vague, generic and undefined God. And the Founders thought it was absolutely necessary to not define God's attributes too specifically or indentify him as the God of the Bible.
The Founders specifically drew a connection between indentifying God's attributes too specifically and denying the rights of those whose religion denied those specific attributes of God. The Founders faced a dilemma: They posited a conception of individual rights which were both universal and antecedent to majority rule. Moreover, they needed to refute prior claims of right such as Divine Right of Kings which explicitly invoked God to justify such claims. The Founders thus needed to tie their notion of rights to God to make such individual rights unalienable.
Yet, invoking God as the source of political claims had long led to terrible persecution itself (indeed, it was the source of religious persecution which they were trying to solve). If it is claimed that God X grants rights, the Founders feared political forces would deny rights to those who didn't believe God X. But such rights, the Founders believed, applied to everyone regardless of what they believed.
Take for instance, Jefferson's Virginia Statute on Religious Freedom which begins "Whereas Almighty God hath created the mind free...." That natural rights document like all of the other natural rights documents and public supplications to the God of the Founding era invokes a generic monotheistic God as the guarantor of natural rights. Jefferson later explains exactly why such a God was not indentified as the Christian God:
Where the preamble declares, that coercion is a departure from the plan of the holy author of our religion, an amendment was proposed, by inserting the word "Jesus Christ," so that it should read, "a departure from the plan of Jesus Christ, the holy author of our religion;" the insertion was rejected by a great majority, in proof that they meant to comprehend, within the mantle of its protection, the Jew and the Gentile, the Christian and Mahometan, the Hindoo, and Infidel of every denomination.
James Madison, who successfully fought for the Bill's passage in Virginia, corroborates Jefferson's account and endorses Jefferson's argument in his Detached Memoranda:
In the course of the opposition to the bill in the House of Delegates, which was warm & strenuous from some of the minority, an experiment was made on the reverence entertained for the name & sactity of the Saviour, by proposing to insert the words "Jesus Christ" after the words "our lord" in the preamble, the object of which, would have been, to imply a restriction of the liberty defined in the Bill, to those professing his religion only. The amendment was discussed, and rejected by a vote of agst (See letter of J. M. to Mr Jefferson dated ) The opponents of the amendment having turned the feeling as well as judgment of the House agst it, by successfully contending that the better proof of reverence for that holy name wd be not to profane it by making it a topic of legisl. discussion, & particularly by making his religion the means of abridging the natural and equal rights of all men, in defiance of his own declaration that his Kingdom was not of this world.
I note all of this because Christian Nationalists are fond of claiming that rights come from God. But they mean rights come from their fundamentalist Christian God and men thus only have the right to do what this God permits in the Bible. This fundamentally misunderstands our Founding principles. But it does contain a half-truth: Our Founders did argue that natural rights come from God. Here is the way one Roy Moore supporter put it in this comment section on World Magazine's blog.
Roy Moore teaches, compellingly, that the founders of America worked from the premise that men are endowed by their Creator with certain inalienable rights. A just government must recognize and protect those rights precisely because they derive from a higher authority whom every human authority is accountable to. When American governments (state and federal) refuse to acknowledge God, they abandon their rationale for identifying and protecting those rights.
Yet, as we have seen, these same Founders never identified God as the God of the Bible or the God that revealed the Ten Commandments to Moses, but rather as an amorphous "Nature's God." Indeed, our freethinking, Theistic Rationalist Founders either disbelieved or strongly doubted that God revealed the Ten Commandments to Moses.
If Roy Moore had simply slapped a ten ton monument of the Declaration of Independence or any one of the Founding Era's natural rights documents with their references to a generic, rights granting Creator, I don't think there would have been a problem (I certainly wouldn't have a problem with that).
By defining, in a public supplication, our rights granting God as the God who revealed the Ten Commandments to Moses, Roy Moore betrayed our Founding Fathers who never did the same.