Frank McCloud (Bogart) once and for all proves he isn't a coward by killing Johnny Rocco (Edward G Robinson) and his gang of murdering mobsters. Frank is shot but radios that he is alive and coming back.
The ending doesn't say if he stays with Nora or not.
Joseph C adds...:
In the final scene of the movie, Nora (Lauren Bacall) answers the phone, has a brief conversation, hangs up and says ""It's Frank! He's alright and he's coming back to us." I think it is pretty clear that Frank and Nora stay together.