During World War II, Bosley served in the United States Navy. While attending DePaul University, in Chicago, in 1947, he made his stage debut in Our Town with the Canterbury Players at the Fine Arts Theatre. Bosley performed at the Woodstock Opera House in Woodstock, Illinois, in 1949 and 1950 alongside Paul Newman.
His first motion picture role was in 1963, as the would-be suitor of Natalie Wood in Love with the Proper Stranger. Other films include The World of Henry Orient, Divorce American Style, Yours, Mine and Ours, Gus and the made-for-television The Triangle Factory Fire Scandal. Bosley shared a heartfelt story about his experience with the Holocaust in the documentary film Paper Clips.
Among his early television appearances was in 1960 on the CBS summer replacement series, Diagnosis: Unknown, with Patrick O'Neal. In 1962, he portrayed Assistant District Attorney Ryan in the episode "The Man Who Wanted to Die" on James Whitmore's ABC legal drama The Law and Mr. Jones. In 1969, he appeared in a comical episode of The Virginian.
Bosley's best known role was the character Howard Cunningham, Richie Cunningham's father, in the long-running sitcom Happy Days. Bosley was also known for portraying Sheriff Amos Tupper on Murder, She Wrote. He also portrayed the eponymous Father Frank Dowling on the TV mystery series, Father Dowling Mysteries. Among myriad television appearances, one notable early performance was in the "Eyes" segment of the 1969 pilot episode of Rod Serling's Night Gallery, directed by Steven Spielberg and starring Joan Crawford.
Also a voice actor due to his resonant, fatherly yet expressive tone, Bosley hosted The General Mills Radio Adventure Theater, a 1977 radio drama series for children.
He voiced many cartoon characters, including Harry Boyle in the animated series Wait Till Your Father Gets Home. He provided the voice of the title character in the 1980s cartoon The World of David the Gnome and the shop owner Mr. Winkle in the children's Christmas special The Tangerine Bear. He also narrated the movie documentary series That's Hollywood. Additionally, he played the narrator B.A.H. Humbug in the Rankin/Bass animated Christmas special The Stingiest Man In Town. Bosley was also the voice of Mister Geppetto, Pinocchio's 'dad' in Filmation's Pinocchio and the Emperor of the Night, released in 1987. Bosley also starred in the 2008 Hallmark Channel television movie Charlie & Me. In 2010 he appeared in The Backup Plan starring Jennifer Lopez.