John Witherspoon was both an orthodox Christian and a philosophical rationalist, a man of the Enlightenment. There is evidence in the historical record of Witherspoon giving orthodox sermons. However, as President of Princeton University (then The College of New Jersey) when he taught James Madison and many other Founders political theory, Witherspoon turned to the teachings of the Scottish Enlightenment, not orthodox Calvinism. And that's because Calvinism spoke little to what the American Founders accomplished politically.
I blogged more about that here.
I must note that the "rationalism" he embraced, even if termed "enlightenment rationalism," Witherspoon thought entirely compatible with and complementary towards revelation. In this sense he was not unlike Aquinas. Witherspoon in his Lectures on Moral Philosophy wrote "[t]here is nothing certain or valuable in moral philosophy, but what is perfectly coincident with the scripture."
Here is how Witherspoon, in those Lectures, explained his view on how reason and revelation were supposed to work together:
If the Scripture is true, the discoveries of reason cannot be contrary to it; and, therefore, it has nothing to fear from that quarter. And as we are certain it can do no evil, so there is a probability that it may do much good. There may be an illustration and confirmation of the inspired writings, from reason and observation, which will greatly add to their beauty and force.
The noble and eminent improvements in natural philosophy, which have been made since the end of the last century, have been far from hurting the interest of religion; on the contrary, they have greatly promoted it, Why should it not be the same with moral philosophy, which is indeed nothing else but the knowledge of human nature?
Further, Witherspoon noted:
I am of opinion, that the whole Scripture is perfectly agreeable to sound philosophy; yet certainly it was never intended to teach us every thing. The political law of the Jews contains many noble principles of equity, and excellent examples to future lawgivers; yet it was so local and peculiar, that certainly it was never intended to be immutable and universal, It would be more just and useful to say that all simple and original discoveries have been the production of Providence, and not the invention of man.
On the whole, it seems reasonable to make moral philosophy, in the sense above explained, a subject of study. And indeed let men think what they will of it, they ought to acquaint themselves with it. They must know what it is, if they mean even to show that it is false.
On the very first page on Lectures, Witherspoon explains what "moral philosophy" is:
MORAL Philosophy is that branch of Science which treats of the principles and laws of Duty or Morals. It is called Philosophy, because it is an inquiry into the nature and grounds of moral obligation by reason, as distinct from revelation.
As noted above, when it came to political theory, there is no evidence that Witherspoon taught his students at Princeton the Bible or Calvinism. Rather he taught them this "moral philosophy" discovered by reason, as distinct from revelation, that was, Witherspoon asserted, ultimately compatible with the scriptures.
But make no mistake, compatibility with the Bible (and it's debatable whether Witherspoon's enlightenment political teachings WERE compatible with scripture) does not mean the Bible was from where Witherspoon's political teachings derived. As historian James McAllister summed it up:
The answer to the question regarding the biblical contribution to Witherspoon's teaching about the law and liberty is: almost nothing ... his theory of society and civil laws was based not on revelation but on the moral sense enlightened by reason and experience.
-- James McAllister, “John Witherspoon: An Academic Advocate for Religious Freedom” in A Miscellany of American Christianity, ed. Stuart Henry (Durham: Duke University Press, 1963), p. 218.