These Wilder Years (1956)

The wealthy yet unfulfilled Steve Bradford (James Cagney) decides it's time to atone for a troubling event. He returns to his old college town, where in his younger days he gave up for adoption a child he fathered out of wedlock. He goes to the local orphanage searching for clues and meets Ann Dempster (Barbara Stanwyck), who runs the agency. Though they are drawn to one another, Ann refuses to release any information to Steve. So out of desperation he takes legal action against her.