Did Batman actually kill Ra's Ah Ghul?

BATMAN.

Batman is one of my favorite superheroes. :) I still haven't been able to see The Dark Knight and such, but I still think he's pretty awesome. He has one of the coolest costumes (With a cape that actually has a purpose. Like, no other superhero has that.) I also like him because he's not perfect. He's not some alien or science experiment. He has a decent purpose for doing what he does, and he makes mistakes.
Even if you don't know much about this guy, you probably know he doesn't kill whatever bad guy he's fighting. Now that is a theory I've never fully understood or agreed with. Sometimes people do stuff that's deserving of death. But that's not what I want to talk about in this post.

Lately I've been thinking about the end of Batman Begins, where him and that Ra's Al Ghul guy (that's like the most complicated name ever)  are on the train. Here's the scene and part I'm talking about here.

Batman says "I won't kill you. But I don't have to save you." And then he "flies" away and that Razzy guy dies in an fiery explosion. It's like there's a difference between killing him and letting him die when he could have done something about it. I don't really get this. Batman could have saved him. But he didn't. Isn't that the same as killing him?
This isn't whether Razzy deserved death. I think he did. Batman should have killed him. I'm talking about if Batman really killed this guy or not. And this type of thing translates into a lot of different things: Is withholding the truth the same as lying? God tells us not to take revenge on our enemies. But if we see them struggling and don't help them even if we could, is that the same as hurting them? Batman could have saved this guy but he didn't.

I think he killed him. I don't know how it could be justified in any other way. No, he didn't take out a knife and stab him or shoot him in the head, but he basically killed him because he didn't do anything to help him live.

I'm not entirely sure though. This has been bugging me for like 3 months. I'd really like to know what you guys think about this particular thing! :D

Bottom Line: Did Batman technically "kill" Razzy, or is it different because he didn't actually kill him, he just didn't save him.

Much love,
Amanda :)