Walter Williams correctly notes that the United States is not a "pure democracy," but rather a representative republic. However, I think he makes a semantical error when he claims that the United States is in no sense a "democracy," and in every sense a "republic."
We often hear the claim that our nation is a democracy. That wasn't the vision of the founders. They saw democracy as another form of tyranny. If we've become a democracy, I guarantee you that the founders would be deeply disappointed by our betrayal of their vision. The founders intended, and laid out the ground rules, for our nation to be a republic.
The word democracy appears nowhere in the Declaration of Independence or the Constitution -- two most fundamental documents of our nation. Instead of a democracy, the Constitution's Article IV, Section 4, guarantees "to every State in this Union a Republican Form of Government."
The United States is a (small l, small d) "liberal democracy"; in fact we were the first true liberal democracy. However, the concept of a representative constitutional republic (a term of which I think Williams would approve), is a specific version of the larger general concept of liberal democracy.
Look, I completely support the notion of a constitutional republic over a pure democracy; in fact, I'm deeply skeptical of majorities and their abilities to enact their whims into legislation and referenda.
However, I have a feeling that when someone generically refers to the US as a "democracy" Williams feels the need to correct them. I think Williams would be wrong to do so. Technically we are a type of democracy, just not a "pure" democracy.