If you're concerned about word choice, you can change the word. "He noiselessly nudges the door open. And enters."
As others have said though, "pushes" is not really forceful in this context. It definitely can be, you can push someone (though I'd use 'shove' to be forceful).
In general though I find "push" to be a fairly neutral word. If you want to push gently, you have "nudge open the door", "eased open the door". If you want to open quickly, or forcefully, you can "slam open the door", or "violently open the door".
So as a TL;DR: "push" is neutral. The part in bold is just fine.