Here's an alternative... what about the possibility of killing off Aunt May instead?
The killing of a sweet, defenseless old woman would be a far bigger
Kick The Dog moment for the villain than killing a girlfriend would. Plus, it frees up the identity of the villain who does the deed as it need not be the Green Goblin.
Meanwhile, it would still achieve the same thing with Captain Stacy's warning - except here, the person who is close to Peter doesn't even know he's Spider-Man yet gets targeted anyway. This would really cause Peter to push away everyone in his life. And since Gwen knows his secret identity and thus why he's doing that, cue the whole "I'm doing this to protect all of you"/"Don't you think we should get to decide for ourselves?" type argument +/- Gwen questioning whether he should simply give up being Spider-Man.
The only thing it doesn't fulfill is the foreshadowing given by Captain Stacy's last request. Though, the promise can be brought; allowing Gwen to bridge into how her father too faced the same dangers and risks because of his job yet did not push everyone in his life away and was able to build a life/family for himself.