In 1952, he was appointed professor of pediatrics at Case Western Reserve University. Robbins was elected a fellow of the American Academy of Arts and Sciences in 1962. From 1966 to 1980, Robbins was dean of the School of Medicine at Case Western. In 1980, he assumed the presidency of the National Academy of Sciences' Institute of Medicine. Five years later, in 1985, Robbins returned to Case Western Reserve as dean emeritus and distinguished university professor emeritus. He continued to be a fixture at the medical school until his death in 2003. The medical school's "Frederick C. Robbins Society" is named in his honor.
He received the Nobel Prize in Physiology or Medicine in 1954 along with John Franklin Enders and Thomas Huckle Weller, making Robbins the only Nobel laureate born in Alabama. The award was for breakthrough work in isolating and growing the polio virus in tissue culture, paving the way for vaccines developed by Jonas Salk and Albert Sabin. He attended school at the University of Missouri and Harvard University.