From what I have researched, North is correct in his essential claim.  Other scholars have noted something similar.  For instance, in this post I noted Thomas Pangle and Cushing Stout, whose work North cites, concluding that there is a connection between the US Constitution's benign approach to religion and the key Founders' enlightened and benign personal religious creed.  Indeed, one could argue, as does Dr. Gregg Frazer, that the Founders' unitarianism or theistic rationalism was the "political theology" of the American Founding.

Ideas have consequences and it was these heterodox unitarian ideas, not orthodox Christianity, that drove the US Founding's approach to religion and government.  However, such heterodoxy or heresy wasn't a popular creed, but rather was disproportionately believed in by the elite Whigs.  Whatever the religion of a majority of the US population (either nominal Protestant Christianity, which itself can tend towards "Deism," or orthodox Protestant Christianity) orthodox Churches held a great deal of institutional power.  With such power, they had to essentially "consent" to the elite Whig's new plan on government.  And they did.  But not all of them.  For instance, the Reformed Presbyterian Church of North America (Covenanters) to whom North dedicates his book.  From the very beginning they "smelled a rat in Philadelphia."  

So the notion that there was a secret coup, a "bait and switch" as Michael Zuckert put it, to "sell" a Christian audience non-authentically Christian ideas is not new.  James Renwick Willson was one of those covenanters who in 1832 made arguments very similar to North's.  And he was burned in effigy for this sermon which called all of the Presidents from Washington to Jackson "infidels" and not more than "unitarians."  I think Willson got at the truth, but did so by shattering a sacred cow -- a social myth.  The kernel of truth that David Barton et al. have is that many folks in the 19th Century did believe in the "Christian America" social myth as a cultural prejudice.  And many of their bogus, unconfirmed quotations source back to 19th Century places that pushed this social myth.  

Now the non-respectable has become the respectable and secular scholars more or less agree with the claims of James Renwick Willson and Gary North that America didn't have an authentically "orthodox Christian" Founding.

Anyway the following are some of North's amusing reactions.  He reacts to the US Constitution the way a hard core theocrat should.

On Jefferson's and Adams' personal religious beliefs:

In their old age, Adams and Jefferson renewed their friendship in a long correspondence that lasted for more than a decade. Their letters reveal that they were almost totally agreed on religion. They hated Christianity, especially Calvinism....After surveying their letters, Cushing Strout concludes: "Whatever their political differences, Jefferson and Adams were virtually at one in their religion." Strout identifies the creed of this religion: unitarianism.  pp. 140-41.

On James Madison:

James Madison was a covenant-breaking genius, and the heart and soul of his genius was his commitment to religious neutralism. He devised a Constitution that for two centuries has fooled even the most perceptive Christian social philosophers of each generation into thinking that Madison was not what he was: a unitarian theocrat whose goal was to snuff out the civil influence of the trinitarian churches whenever they did not support his brainchild. For two centuries, his demonic plan has worked. pp. 374-75.

On George Washington and his avoidance of communion:

Here was the strange situation: George Washington was formally a communicant church member who systematically refused to take communion. The institutional problem here was the unwillingness of church authorities to apply formal church sanctions. Any church member who refuses to take communion has thereby excommunicated himself. A refusal to take communion or a prohibition against one's taking communion is what excommunication means. Self-excommunication is excommunication, just as surely as suicide is first-degree murder. Nevertheless, the churches to which Washington belonged did not take official action against him by either requiring him to take communion or by publicly excommunicating him. It was this disciplinary failure on the part of these churches that led to the public legitimizing of Washington as a Christian. This failure later indirectly legitimized the Constitution that he conspired to impose on the nation. Without Washington's support of the actions of the Convention, the Constitution would never have been ratified. But Washington was deemed either too powerful or too sacrosanct to bring under church discipline.  pp. 160-61.