No Way Out (movie, 1950)

The Biddle brothers, shot while robbing a gas station, are taken to the prison ward of the County Hospital; Ray Biddle, a rabid racist, wants no treatment from black resident Dr. Luther Brooks. When brother John dies while Luther tries to save him, Ray is certain it's murder and becomes obsessed with vengeance. But there are black racists around too, and the situation slides rapidly toward violence.