To him who believes in the Existence and Attributes physical and moral of a God, there can be no obscurity or perplexity in defining the Law of Nature to be his wise benign and all powerful Will, discovered by Reason.
– John Adams to Thomas Boylston Adams, March 19, 1794. Adams Papers (microfilm), reel 377, Library of Congress. Seen in James H. Hutson’s, “The Founders on Religion,” p. 132.
The strongest argument for letting the "under God" in the pledge pass constitutional muster is that natural rights Founding era documents invoke such a God, and as such, making a public recitation of the Declaration of Independence unconstitutional would yield perverse results. [And the other side properly notes making a group of people "pledge allegiance" to a God has a coercive element that would be missing from mere government endorsement of the theistic Declaration of Independence.]
Kevin Hasson, founder and president of the Becket Fund for Religious Liberty, whom we've seen debate Michael Newdow, makes what I think a very good historical and philosophical argument for his side. And that is the God of natural rights is the God of nature, not the God of the Bible or not necessarily the God of the Bible. America's Founders believed in granting religious freedom for all. And they reasoned, you couldn't have religious freedom if the natural rights source for that unalienable right was a sectarian (i.e., "Christian") deity. Hence the invocation of a generic, non-sectarian deity: Nature's God.
The term "nature" as was used during America's Founding era (and still today) meant discoverable by reason as opposed to revealed by scripture and as such, "Nature's God" is God insofar as we can discern His existence and determine His attributes from reason unassisted by revelation. (See the above Adams' quotation.)
It's amusing to see one of Roy Moore's cronies struggle with this conception. He quotes the New York Sun article:
[T]he God in the pledge is the same God referred to in the Declaration of Independence, but is not the deity in the Bible. “It wasn’t the Christian God. It wasn’t the Jewish God. It was the philosopher’s God,” Mr. Hasson said. He said the “under God” reference refers to a creator early philosophers and scientists like Aristotle concluded “could be known by reason alone.”
And he reacts:
Mr. Hasson has the honor of making an argument that I can unequivocally state I have never heard in any previous Establishment Clause case. He is actually claiming that the God referenced in the Pledge is not the God of the Bible, but rather is some amorphous “philosopher’s God.”
That Mr. Jones has never heard of this argument before simply shows that he hasn't done his homework. He either hasn't studied the historical documents on this issue, or if he has, his analysis is confused by the specious arguments put forth by the likes of Roy Moore.
It would be a mistake, however, to try to exclude the Biblical God from this conception. Rather, a prime reason why America's Founders turned to "nature" and not "scripture" to ground America's public creed was to be inclusive. Natural theology was a lingua franca (a link language) in which orthodox trinitarians, unitarians, theistic rationalists and deists all could speak.
For a good place to learn how the Founding era viewed the concept of "natural religion" (that is what man can discern about God's universe through reason unassisted by scripture), google the terms "natural religion" and "Dudleian Lectures." As will be seen, some orthodox trinitarian Christians did promote the concept of "natural religion," for instance Samuel Langdon, John Witherspoon, and many others. But when they indulged in this theology, note, they stayed true to its method, which was, again, what man discovers from reason unassisted by revelation. When orthodox Christians indulged in natural theology, they found reason and revelation perfectly agreed. Deists, on the other hand, found that natural theology didn't at all agree with what is revealed in scripture. And theistic rationalist/unitarians found that sometimes reason and revelation agreed, sometimes they didn't.
But in the end, natural theology is defined as what man discovers from reason, and when it came time to declare independence, America turned to the laws of nature and nature's God, not what is revealed in scripture.