I haven't read this entire piece, but this looks to be a very informative piece on the public or "civil" religion of America [Hat tip: Mollie at Get Religion].
It is crucial that presidents never mention "Jesus" or "Christ" in this context, which would cross the line from civil religion into sacral religion. American civil religion transcends denomination and religious affiliation. The "Almighty" of civil religion could be Christian, Jewish or Muslim. Even Wiccans might feel kinship with Jefferson's "Nature's God." Strict atheists, however, would be alienated.
As the piece seems to insinuate, the "Almighty" of the civil religion could be a very unorthodox Providence. Indeed, our first four Presidents who commonly acknowledged such a God in their public pronouncements, were heterodox (theological) unitarians (certainly Jefferson and Adams were, Madison and Washington likely were).
So what then is Rousseau's "civil religion"? This doctrine is a form of "deism" and I think it is what the Supreme Court refers to when they apply their doctrine of "Ceremonial Deism."
Now, what this does not mean is that it's "okay" for government to endorse the concept of "Deism," which posits a cold, distant non-interventionist God, but "not okay" for government to endorse any other religious notion. This is a mistake that some people make (see for instance, Dennis Teti making this mistake). As I have noted before, the term "deism" as used during the Founding sometimes referred to generic monotheism. So the "ceremonial deism" of the "civil religion" really refers to a Lowest Common Denominator form of monotheism. As Larry Arnhart puts it in this post:
As I read Tocqueville, he is applying to America Rousseau's idea of "civil religion," in which the only required doctrines are the existence of a providential God who enforces a moral law by punishing the bad and rewarding the good. Is this the doctrinal content of the morally healthy religion?
Justice Scalia's very interesting dissent in McCreary explores this notion of a Lowest Common Denominator God. However, he ultimately misses when he asserts that Moses divinely receiving the Ten Commandments is part of the civil religion; they are not. The heterodox unitarianism to which the key founders (for example, the above mentioned first four Presidents) adhered, which viewed Man's Reason as opposed to Biblical Revelation as the ultimate discerner of Truth, was highly dubious of the notion that Moses divinely received the Ten Commandments from God.