This is such a frustratingly ill-informed argument that the other side needs a simple link to address it. So here it is —remember it:
Yes, it is true, the Framers meant to establish “a Republic.”
And yes, they openly and repeatedly criticized “democracy.”
But the “democracy” they were criticizing was “direct democracy,” and the “Republic” they were championing was “representative democracy.”
So can you guys (and it’s almost ALWAYS guys) please just give up on this silly “I’m-so-much-smarter-than-you high school debater’s quibble:
Yes, we are a Republic,
Yes, we are a “representative democracy,”
which means (on the logic of “a Ford truck is a truck”)
Yes we are “a democracy.”
We are just not a direct democracy—and I don’t know anyone who ever said that we are (and I’d be happy to join with anyone to argue we should not be).
 Here are just some of the relevant quotes—and give it up or I’ll have to collect more—from The Federalist Papers:
The Federalist №10 (James Madison), 57 (“A republic, by which I mean a government in which the scheme of representation takes place”);
The Federalist №14 (James Madison), 77 (“The true distinction between these forms was also adverted to on a former occasion. It is, that in a democracy, the people meet and exercise the government in person; in a republic, they assemble and administer it by their representatives and agents”);
The Federalist №39 (James Madison), 233–34 (“[W]e may define a republic to be, or at least may bestow that name on, a government which derives all its powers directly or indirectly from the great body of the people, and is administered by persons holding their offices during pleasure, for a limited period, or during good behavior. It is essential to such a government that it be derived from the great body of the society, not from an inconsiderable proportion, or a favored class of it; otherwise a handful of tyrannical nobles, exercising their oppressions by a delegation of their powers, might aspire to the rank of republicans, and claim for their government the honorable title of republic. It is sufficient for such a government that the persons administering it be appointed, either directly or indirectly, by the people; and that they hold their appointments by either of the tenures just specified”).
The great (and really, read his fantastic book, Founding Finance; Whiskey Rebellion is at the top of the “read next” list) William Hogeland has written a response to the above that convinces me I’m not being clear.
Hogeland criticizes my characterization of “democracy.” He accepts that some—especially Madison—spoke against direct democracy. He asserts that others spoke against a representative democracy that was “too representative.” And he claims (importantly) that there were many praising the democratic aspects of the traditions and constitution. As he writes,
Lessig’s claim that when the framers criticized democracy they meant only to criticize direct democracy — holding a popular referendum on every law and issue, with no representative layer — doesn’t stand up to a second’s scrutiny.
The key disconnect in that sentence is “only”: I certainly didn’t meant to say—and I don’t think that in fact I do say—that when “the framers criticized democracy they meant only to criticize direct democracy.” Because the point of the above is not to describe their conception of “democracy”; it is to describe their conception of “republic.” Whatever they said about democracy—and Hogeland accepts they did criticize direct democracy—my point is what they meant by “a Republic.” And that is, again, that they meant “a Republic” to involve a “representative democracy.” No doubt, they had high (as in elite) hopes for that at the start. Very soon into the experiment, they saw Madison’s predictions about cream rising to the top were false. But regardless of how compelling the democratic part was, my only point is that there was a democratic part to the idea of “a Republic”—which is why, again, the right-wing trope, “The United States is not a ‘democracy,’ it is ‘a Republic’”—is so wrong.