The U.S. was originally set up as a non-democracy or partial democracy. Over time, it has evolved into something that is, theoretically, a democracy. Since you worded your question in the present tense, the answer is yes.
The only way that anyone can pretend otherwise, and we have seen both of those here, is by:
1) Pretending that nothing has changed since 1789; or
2) Misdefining "democracy" in overly-narrow terms so that the term is restricted to a direct democracy or one without any restraints of the nature of the Bill of Rights.
The claim that there is an inherent contradiction between the concepts of "democracy" and "republic" is a false assertion. There is no such contradiction. A republic is not necessarily a democracy, but it can be one, and the U.S. is supposed to be one.
A republic that is not a democracy is an aristocracy instead: rule by the few through the mechanics of a republic. The Roman Republic and the Union of Soviet Socialist Republics were both non-democratic republics. However, the U.S. is not like either of those. We are (or are supposed to be) a democratic republic -- and hence, a democracy.