Well, ok, but we are a democratic republic.
All of our political chief executives are chosen by the popular vote, as are all of our legislators and many of our judges. We also have direct ballot initiative processes in about 23 states. We are not a democracy in the Athenian sense, of course, but there's no particular reason to cede to the ancient Greeks the definition of "democracy". Our system was democratic even before it was a republic. Take a look at American colonial governance. It was representative in some parts, town meeting style in New England, and quite democratic.
And our republic doesn't look much like the Roman Republic, where class limitations applied to what offices one could hold.
We aren't a classical democracy, and we're not a classical republic. We're a democratic republic, or a republican democracy, and always have been. really. There aren't any historical precedents for what we've done here: we broke the mold.
The Roman Empire, you meant that surely.