Neutral evil?

It's an anti-hero. He seems to be nostalgic (i say that because he surely lost his family). 

I am pretty sure that if the military harms him they will have a payback.

I think it's a bad idea to think of this Godzilla as good or evil. He's an animal, and they don't really think or behave in terms of morality.

In an interview Gareth Edwards he said he made a point to have Godzilla never look at a human being because we're so small to him he doesn't even perceive us. If this is the case, then he can't intentionally hurt that which he doesn't even know to exist.