Prosky, a Polish American, was born Robert Joseph Porzuczek in the Manayunk neighborhood of Philadelphia, Pennsylvania, to Joseph and Helen Porzuczek. His father was a grocer and butcher. He was raised in a working-class neighborhood and studied at the American Theatre Wing, later graduating from Temple University. He performed at Old Academy Players, a small theater in the East Falls section of Philadelphia, adjacent to Manayunk.
Prosky appeared in Thief, Christine, Hanky Panky, The Natural, Hoffa, Broadcast News, Things Change, The Great Outdoors, Loose Cannons, Rudy, Mrs. Doubtfire, Green Card, and Dead Man Walking. In addition to appearing in films, Prosky appeared as a regular on Hill Street Blues, Danny, and Veronica's Closet. He was considered for the role of Martin Crane in Frasier and later made a guest appearance in the series as a reclusive writer. He portrayed Rebecca Howe's father on Cheers. Prosky also portrayed Jake "the Snake" Connolly on a 2-part 1991 episode of Coach.
He also appeared as Will Darnell, the man who owned the auto repair shop where Arnie Cunningham (Keith Gordon) rebuilds the possessed car "Christine" in the film John Carpenter's Christine based on Stephen King's novel. Prior to his film and television career, he appeared in numerous productions at the Arena Stage in Washington, D.C., most notably as Willy Loman in Death of a Salesman. Prosky often performed at Arena Stage with over 100 stage credits to his name at that theatre alone. He also originated the role of Shelly Levene in David Mamet's Pulitzer Prize-winning play Glengarry Glen Ross. He was also a board member of the Cape May Stage in Cape May, New Jersey.
In 1960, he married Ida Hove, who survived him, along with three sons, Stefan, John and Andrew Prosky. Andrew and John Prosky are also actors.
On December 8, 2008, Prosky died at age 77, 5 days before his 78th birthday, from complications following a heart procedure. At the time of his death, he was living in the Capitol Hill neighborhood of Washington, D.C.