The Constitutional Republic is designed to protect the rights and freedoms of the individual via the Constitution and bill of rights. If by Democracy one is referring solely to majority rule, then protection against individual rights and freedom are lost.
The U.S. should fear the latter, not the former.
Saying "the US isn't a democracy, it's a constitutional republic", is like saying "Garfield isn't a cat, he's a cartoon and a vertebrate". None of these things are mutually exclusive.
Democracy: government where political power ultimately comes from the popular vote. The opposite of democracy is autocracy.
Republic: government where the state belongs to the people and is run for their benefit. The opposite of republic is monarchy.
Constitutional: government with ground rules for how the levers of power work that nobody is allowed to break. The opposite of constitutional government is absolute government.
You can have government on any of these axes. Being a constitutional republic doesn't preclude being a democracy. Contrast the United States and North Korea. Both are constitutional republics (fundamental law, no monarch), but one is democratic (all votes matter), and the other is autocratic (only one vote matters).
Democracy does NOT mean "absolute rule by a mere majority". Republicans lie and weasel to try to get people to think this so that they can say that democracy is bad and we should accept minority rule instead (aka, autocracy)
Monarchy: yes king
Republic: no king
Democracy: yes votes
Autocracy: no votes
Constitutional: yes constitution
Absolute: no constitution
You can have these in any combination.