This one is primarily promulgated by Peter Marshall, whose historiography is abominable.
Part of the myth involves conflating America's two foundings -- the earlier colonial founding of Virginia, Connecticut, Massachusetts, etc. -- with the Founding of the federal government from 1776-1789. (This helps to peddle the "Christian Nation" myth generally.) To distinguish between the two, Michael Zuckert suggests we use different terms to describe these two "foundings." The federal Founding Fathers were Washington, Adams, Jefferson, Madison, Franklin, et al. Men like John Winthrop, John Smith, Thomas Hooker, and Roger Williams were planting fathers.
Whereas the Federal Founding documents are arguably secular (or perhaps generally theistic), earlier colonial charters of the planting fathers used explicitly biblical language and otherwise covenanted with the Triune Christian God (save for Roger Williams' Rhode Island). Therefore, "Christian America" proponents try to find some explicit connection between the planting and Founding Fathers to show they were of one vision.
Peter Marshall's myth is that George Washington, as leader of the Constitutional Convention, handed out the Fundamental Orders of Connecticut for the delegates to use as a model. As he put it:
George Washington, who served as president of the Constitutional Convention, ordered that that every delegate have a copy of Connecticut’s Constitution. He did so, said Marshall, “because it was so powerfully done, so rooted in Holy Scripture, in the Word of God, such an effective document, [that] Washington wanted that to be a reference work for the federal Constitution work they were about to get into.”
Nothing in the text of the Constitution, the Federalist Papers or the hundreds of pages of Madison's notes on the Constitutional Convention and of the others who kept notes, shows this. (Or, if I missed something, please let me know.)
The kernel of truth in the myth is that the colonial orders did anticipate some of the ideas of representative self government that the Founding Fathers would later implement. The Fundamental Orders of Connecticut was in some sense, an early example of an experiment with self-government. Perhaps some of what the Founders thought to be novel ideas in their Novus Ordo Seclorum, turned out to be not so new after all.
However, whatever useful ideas the Founding Fathers took from the earlier colonial charters were secular. When comparing the language in the earlier colonial charters to that of the US Constitution what's striking is just how different their approaches are to religion and government. The US Constitution completely and utterly lacks explicitly biblical language or a covenant to the God of the Bible, but instead imposes a religiously neutral "no religious test" clause in Article VI, Clause 3. This language is 180 degrees from the preamble of the FOC which states:
For as much as it hath pleased Almighty God by the wise disposition of his divine providence so to order and dispose of things that we the Inhabitants and Residents of Windsor, Hartford and Wethersfield...well knowing where a people are gathered together the word of God requires that to maintain the peace and union of such a people there should be an orderly and decent Government established according to God...to maintain and preserve the liberty and purity of the Gospel of our Lord Jesus which we now profess, as also, the discipline of the Churches, which according to the truth of the said Gospel is now practiced amongst us; as also in our civil affairs to be guided and governed according to such Laws, Rules, Orders and Decrees as shall be made, ordered, and decreed as followeth:
Similarly look at the language of the Mayflower Compact:
”In the name of God, amen. We, whose names are underwritten . . . having undertaken for the glory of God, and advancement of the Christian faith, and the honor of our king and country, a voyage to plant the first colony in the northern Parts of Virginia; do by these presents, solemnly and mutually, in the presence of God and one another, covenant and combine ourselves together into a civil body politick, for our better ordering and preservation, and furtherance of the Ends aforesaid.”
Now compare those with the preamble to the US Constitution:
”We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this Constitution for the United States of America.”
As Dr. David Mazel put it:
The [Mayflower Compact] gives us a crystal-clear example of how a charter is worded by people deliberately founding a Christian polity. We are told directly that the colony is being “undertaken for the glory of God, and advancement of the Christian faith.” The Founding Fathers could have used similar wording, but didn’t. The rationales for creating the Union is purely secular: insuring tranquility, providing for defense, promoting the general welfare, and securing the blessings of liberty.
Moreover, the language in the Massachusetts Body of Liberties is the very opposite of the First Amendment and Article VI:
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.
As noted, the Founders may well have borrowed some ideas from the planting fathers of Massachusetts, Connecticut, Virginia, etc. But what they borrowed was by in large secular. The explicit language in their documents that established these colonies as "Christian Commonwealths" is entirely missing from our federal founding era documents. It's "covenant theology" with the very hearts -- the covenants to the Triune God -- ripped out.
On religion and government, if the Founding Fathers followed any of the planting fathers' models, it was Roger Williams' Rhode Island, the man who coined the term "wall of separation" between Church and State. And whose government was in principle a secular entity, not founded on a covenant to the God of the Bible.