I'd argue that one actually. Bruce dedicates his life to stopping crime, he has no actual reason for doing this, is not government elected, nor does he have any social contract to keep doing what he's doing.

He is no more responsible for Joker's murders than any other bystander in Gotham, the only difference is he's a bystander and chooses to go out and limit the damage Joker is capable of doing and so what he is willing or even able to do (since Bruce feels that if he starts killing he won't be able to stop himself) is entirely his own philanthropic activities in Gotham. As such, any action he does on his own volition in prevention of crime can be seen in a heroic light. Now would it be better for all involved if Batman killed the Joker? Yes. But if he's right about himself and he does go all Punisher afterwards is that within the realm of respectable vigilantism? Then we get into opinions on the death penalty itself, and so forth.

And I think I had a point here that I was building up to. Ugh, that's what happens when I type when I'm about to go to sleep. Maybe this'll come to me tomorrow.