Hehe I love this nickname! I think it's sweet and adorable. As far as being a full name, that wouldn't be a good idea because it could get dull after a while. I think it's best because in case they don't like "Gabe" they can go by their full name and vice-versa.
My boyfriend's name is Gabriel, I call him Gabe for short. I think it sounds so sweet and gentle, although I would not opt to name my child this as a name, only as a nickname. To be honest, it is a lot better and not so frilly and feminine like Gabby - ugh.