Stitt is best known for his play The Runner Stumbles which ran for nearly 400 performances on Broadway. The work, about a priest in a small town who is eventually implicated in a nun's death, was later adapted into a feature film, directed by Stanley Kramer and starring Dick Van Dyke and Kathleen Quinlan. TACT did an Off-Broadway revival of the work in 2007, and it has had numerous regional productions over the years.

Stitt's other plays included Back in the Race, Labor Day, and Places We Lived. He also wrote the popular 1981 television movie The Gentleman Bandit, starring Ralph Waite and Estelle Parsons.

In addition to his work as a playwright, Stitt was the former executive director of Circle Repertory Company, he worked at such theaters at the American Shakespeare Festival and the Long Wharf Theatre, and taught at the Yale School of Drama, Princeton University, NYU, and Carnegie Mellon University.