That's not what pushed her to do it. There's more to this than a plot summary. The gist of it was that she felt he was the only person she could be herself with, and once they started dating she lost that. She had massive problems with her self-image, basically thinking she's chronically insincere and incapable of real bonds, and Sunao was the exception that gave her hope. The bullying was just the spark, not the deep cause.
Him saying he's also to blame probably just comes from a place of empathy. The guy is pretty emotionally mature for his age (thus something of a chick magnet), so he gets what she went through and feels he should have done more, even if it's a lot to expect from a kid. That's not him "gaslighting himself", that's him being kind. You know. His dominant character trait and all?