There I quoted political scientist Francis Fukuyama from a CSPAN interview about a book he wrote on the matter:
Now, by the French Revolution, we don’t mean just the limited historical event; what we mean is the emergence of what we understand as modern liberal democracy because in the French Revolution, ultimately what it was about was a revolution in favor of the principles of liberty and equality. Now you could substitute the American Revolution for that because, I think in that kind of ideological sense, those two revolutions were equivalent. I mean, they were both revolutions to create what I earlier defined as a liberal democracy as a political system based on popular sovereignty with guarantees of individual rights.
These "liberal democratic" notions written of in America's Declaration of Independence and France's Declaration(s) of the Rights of man are at the heart (soul?) of "constitutional republicanism." (Of course, the East Coast Straussians argue we need the FORM of constitutional republicanism without its "rights oriented" HEART; but that's a story for another day.)
The difference between today's insistence on freedom and equality and that of the French and American Revolutions is that back then they attached metaphysical or religious essences to liberal democracy, whereas today we tend not to; Fukuyama recognized this in "The End of History and the Last Man" where he noted it was History, NOT "natural right" OR the Bible that would eventually vindicate the universal acceptance of liberal democracy, i.e., the notion that man is born free and equal with "rights" against governments, and governments, consequently, derive their legitimacy from the consent of the governed.
For the non-historicist reading of the eventual universality of liberal democracy, see Ellis Sandoz's political sermons where we see "patriotic preachers" -- some of them unitarian, some of them trinitarian, all of them "enlightened," meaning they injected excessive use of man's "reasoning" into their sermons -- arguing the American and French Revolutions as sister events. AND they thought liberal democracy would terminate with the success of the French Revolution which would usher in a "millennial republic" of liberty, equality and fraternity.
Of course, not all who supported the American Revolution supported the French from the start. Edmund Burke and John Adams didn't and they were outliers. However, the view that the two revolutions were sister events dominated American thought (and much of British Whig thought) at the beginning of the French Revolution.
See the following past posts of mine for more in depth examination of these sermons here, here, here, here and here.