I try not to beg anyone to stay, even if I desperately want them to, because I try to respect their right to leave [even if I hate it to the point of feeling ill].
The fact that he let her leave with no trouble isn't proof that he didn't care for her. She made the decision to walk out, and she can't blame him for it [if she wanted him to beg then she obviously cared for him, and he didn't drive her away].
It's her fault. If he wanted her badly enough, he probably should have tried to stop her... but if someone wants to leave you just to see if you keep them, they shouldn't expect =anything= out of you.