One major error of the "Christian America" crowd is to conflate America's Planting (that is, its old colonial order from the 17th Century) with its Founding (1776-1800). In reality, the principles of the two are in great tension and the ideals of the Founding -- the Novus Ordo Seclorum -- represented a great break with past tradition. Dr. David Mazel made this point in this past post by comparing the wording of the Mayflower Compact with that of the US Constitution.
We can also use John Adams himself to illustrate the point. From Massachusetts, Adams, some believe, was a Puritan. Though he had Puritan roots, he was a clearly a different animal than the Puritans who landed on Plymouth rock. For one, they were hard core theocrats and he wasn't. Secondly, they would have executed Adams for "high handed blasphemy."
The Puritan's Massachusetts Body of Liberties (1641) -- what a Christian civil code looked like before the Enlightenment and Church & State were separated -- reads like an American Talibanic code. Here are some examples of their capital laws:
94. Capitall Laws.
(Deut. 13. 6, 10. Deut. 17. 2, 6. Ex. 22.20)
If any man after legall conviction shall have or worship any other god, but the lord god, he shall be put to death.
(Ex. 22. 18. Lev. 20. 27. Dut. 18. 10.)
If any man or woeman be a witch, (that is hath or consulteth with a familiar spirit,) They shall be put to death.
(Lev. 24. 15,16.)
If any person shall Blaspheme the name of god, the father, Sonne or Holie Ghost, with direct, expresse, presumptuous or high handed blasphemie, or shall curse god in the like manner, he shall be put to death.
And here is evidence of John Adams' "high handed blasphemie" that a Puritan prosecutor would be able to use to put Adams to death just as Servetus was in Calvin's Geneva for publicly proclaiming his unitarian heresy.
"The Trinity was carried in a general council by one vote against a quaternity; the Virgin Mary lost an equality with the Father, Son, and Spirit only by a single suffrage."
-- John Adams to Benjamin Rush, June 12, 1812.
"An incarnate God!!! An eternal, self-existent, omnipresent omniscient Author of this stupendous Universe, suffering on a Cross!!! My Soul starts with horror, at the Idea, and it has stupified the Christian World. It has been the Source of almost all of the Corruptions of Christianity."
-- John Adams to John Quincy Adams, March 28, 1816
"If I understand the Doctrine, it is, that if God the first second or third or all three together are united with or in a Man, the whole Animal becomes a God and his Mother is the Mother of God.
"It grieves me: it shocks me to write in this stile upon a subject the most adorable that any finite Intelligence can contemplate or embrace: but if ever Mankind are to be superior to the Brutes, sacerdotal Impostures must be exposed."
-- John Adams to Francis van der Kemp, October 23, 1816.