Julio Frenk, a public health expert who has led the University of Miami since 2015, was named on Wednesday as the next permanent chancellor of the University of California, Los Angeles. He will oversee an elite public institution still reeling from intense protests and a violent attack on student demonstrators that occurred this spring.

Dr. Frenk will become the first Latino to lead U.C.L.A., whose student body is one of American higher education’s most diverse. He will succeed Gene Block,…