I don't think you can do one without doing the other. I'm sure you could do it for lots of other stories, but The Yellow Wallpaper is about a depressed woman who's driven mad by her doctor husband when she's locked in her room as part of a misguided medical procedure. If I were a professor reading a paper on this story that didn't mention gender (at least to the extent of mentioning why the author doesn't think it's significant), I might worry that my student had missed the gendered aspects entirely.
There are lots of different ways to structure a paper like that. I'm pretty sure I could write an acceptable one without mentioning gender at all, but I could also write one bringing in the 19th Century fad of performing hysterectomies on women to "cure" hysteria, or the fact that madness, either congenital as in
The Fall of the House of Usher, or brought on by the ever popular "brain fever", popped up in so many Gothic stories. I could also do one in convoluted run-on sentences like that.
I once wrote an entire paper leaving out the vowel "i", just to see if it was possible. (e.g. "Once was done a whole essay, sans the vowel of 'oneness', just for laughs.")