D. Nelson

I can't think of anything suitable. A quick search I did couldn't find anything either, apart from the examples I mentioned and similar ones using synonyms for gently (e.g. softly).

Maybe someone will be able to suggest something, but I'm not sure that any word like that exists.

I don't think the word you're seeking exists. "Gently close the door", or "pull the door to" are the only natural-sounding ways I can think of to say what you want.