Bellamy was born in Chicago, Illinois. He began his acting career on stage, and by 1927 owned his own theatre company. In 1931 he made his film debut and worked constantly throughout the thirties, establishing himself as a capable supporting actor. He received an Academy Award nomination for Best Supporting Actor for his role in The Awful Truth (1937) and played a similar role in His Girl Friday (1940). He played detective Ellery Queen in a few films during the forties, but as his film career had not progressed, he returned to the stage, where he continued to perform throughout the fifties.

Highly regarded within the industry, he served four years as President of Actor's Equity.

He appeared in Sunrise at Campobello (1960) and Rosemary's Baby (1968) before turning to television during the seventies. An Emmy Award nomination for the mini-series The Winds of War (1983), and a role as a conniving billionaire alongside Don Ameche in Trading Places (also 1983) brought him back into the limelight.

In 1984 he was presented with a Life Achievement Award from the Screen Actors Guild, and in 1987 received an Honorary Academy Award "for his unique artistry and his distinguished service to the profession of acting."

He continued working regularly and gave his final performance in Pretty Woman (1990).

He died as a result of a lung ailment in Santa Monica, California at the age of 87, and was buried in Forest Lawn - Hollywood Hills Cemetery in Los Angeles.

Bellamy has a star on the Hollywood Walk of Fame at 6542 Hollywood Boulevard.