It's easy to say "well he has no legal right, it's all the State's fault!", but Batman -knows- this. This isn't the first time it's happened, or even the tenth. Batman -knows- that every time he captures Joker and turns him over to the authorities, that he will get out and kill again. He has the capability of stopping it. But he chooses not to, -knowing- that that choice will lead to innocent deaths. How that makes him anything but responsible is beyond me. Sure, he's technically free of guilt due to the way the system works, but fact is those people are still in the ground. The result is innocent death, period. And if it was your loved one you had to mourn, you might be a little less apt to buy the "but it's illegal!" excuse for failing to get the job done.
And sure, "serial killer" Frank from the Ellis MAX stuff might be shallow and about self-gratification, but it wasn't always like that. In the 70s and 80s, mainline Marvel Punisher was a lot different. There was no "self-gratification" in anything Castle ever did. He was a dead man walking, living only for the mission, taking what he learned in war and applying it to combatting the criminal underworld. In those days, even calling him by his name would get a reply of "Frank Castle died with his family. I'm the Punisher". The man only slept and ate because he physically had to. He took pleasure in nothing, doing his best to live like a Terminator with his mission the only priority. And as was illustrated more than once, he was driven by self-destruction far more than self-gratification. He was consumed by survivor's guilt, and praying for death in combat to reunite him with his loved ones. He was just duty-bound to fight his best, and that kept the death he wanted from him.
His view on the whole legal-vs-moral debate was abundantly clear and exposited quite regularly in his encounters with heroes. He viewed "no-kill" vigilantes as weak and ineffectual, and in no small part responsible for perpetuating the failures of the criminal justice system. To him, legal justifications were mere words, excuses. And given the artificiality of the legal and governmental system in the first place, he's correct. Results matter. Actions that satisfy the letter of the law but lead to more innocent deaths are not just, period. And being unwilling to make the hard choices and take the heat for doing the right thing when lives are at stake is unforgivable.
Really, it comes down to a simple formula:
Choice A leads to a dead criminal and a hero facing the ire of the police, but a would-be victim that is still alive.
Choice B leads to a free criminal and a dead victim, but the hero is safe from being hunted by the cops.
To the Punisher and those who agree with his view, choice A is the only moral one. Choice B is cowardice. If a hero is willing to jump in front of a bullet to save a life, why would he be unwilling to make a target of himself to save that same life? It's the same thing.
In the Nuremburg trials, we established that human life is a higher obligation than laws or orders, and that there is a moral obligation to break the law and the orders of legal authorities if following those laws and orders leads to innocent deaths.
I'm not seeing how that principle doesn't apply here.