CLEVELAND—Democrat Hillary Clinton, in her first extended response to the massacre in Orlando, denounced “inflammatory anti-Muslim rhetoric” and pushed for stricter gun laws, including the reinstatement of a ban on the sort of assault weapons used by the Florida gunman.

“If the FBI is watching you for suspected terrorists links, you shouldn’t be able to just go buy a gun,” she said here in a reference to two prior anti-terrorism investigations of Omar Mateen, the shooter who was killed Sunday by police. Those probes were eventually...