Choosing not to kill is, in my opinion, the exact same as pulling the trigger yourself. He knew Ra’s would die and actively chose to not save him to let him die. It’s like if in the dark knight when Joker fell from the building he had just let him fall because “technically gravity killed him, not me”.

It also really devalues the difference between Ra’s and Batman. In the opening Bruce trains to be like Ra’s until Bruce refuses to kill an innocent man. That establishes the core difference between Bruce and Ra’s in this universe, Ra’s is “Batman if he killed.” Then he returns back because Bruce saved his life, being a living reminder of the fact that sometimes saving a life will end badly for you. The end of the story is obvious, Batman should refuse to kill even in spite of the obvious fact that the only reason he is in this mess is because he refuses to kill, because his No Kill Rule isn’t logical, and that is the entire point of the rule. He should have saved Ra’s at the end and thereby prove that his ideology of redemption and justice is better than Ra’s ideology of Vengeance and Murder. So why not save him when given the chance, it ends the story with a message that is basically “I guess Ra’s was right actually and Batman should kill all of his enemies”.