No, of course a specific religion wasn’t specified. Adams was a devout man, but he also wanted the DOI to be about the rights of ALL men, not just the colonists. He thought slavery was an affront to the religious principles in which he believed.
His morality extended from his personal faith in Providence to all others.
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Religion is well supported; of various kinds, indeed,
but all good enough; all sufficient to preserve peace and order: or if a sect arises, whose tenets would subvert morals, good sense has fair play, and reasons and laughs it out of doors, without suffering the state to be troubled with it. They do not hang more malefactors than we do. They (i.e. Pennsylvania and New York) are not more disturbed with religious dissensions. On the contrary, their harmony is unparalleled, and can be ascribed to nothing but their unbounded tolerance, because there is no other circumstance in which they differ from every nation on earth. They have made the happy discovery, that
the way to silence religious disputes, is to take no notice of them. Let us too give this experiment fair play, and get rid, while we may, of those tyrannical laws."
Source: Thomas Jefferson, "Religion" in Notes on the State of Virginia (1782), p. 287.