The heady romance between two people caught up in a war, when both are free and dealing with real life issues when only one is free is the core of the movie.
Rick could look at Ilsa and see if she stayed with him, she would always question if that was the right move.
Living with regret, no matter what one gains is a bad bargain. Rick has been around long enough to know this, so he settles it for her.
An act of both love and release.