According to activist 20th century courts, but certainly not according to the debates when the 14th amendment was ratified and certainly not according to court decisions in the first several decades after the 14th amendment was passed.
Nonetheless, it is settled law. I do believe that it happens to be good law.
The Bill of Rights has to extend to the states, otherwise it is meaningless. Prohibiting the Federal Government from banning handguns is meaningless if the states are allowed to ban handguns. Prohibiting the Federal Government from shutting down oppostition newspapers is meaningless if the states can shut down opposition newspapers.
Either these are rights retained by the people, or they are not. If they are retained by the people, the state can no more infringe upon them than the feds.