I get moments like this too, social media can ruin a relationship!
My husband to be works in a place full of inappropriate immature girls, some always have moments where they write stuff to him on Facebook and can be a bit flirtatious, because he'd never talk about it, it used to cause massive arguments. So we sat each other down and promised to tell each other everything. Recently he had a girl add him on Facebook and he's ignored it because I know her from somewhere else and don't trust her. I would say it's probably a good thing your fiance has ignored the friend request, if there was anything going on surely he would of accepted her? I think don't make any rash decisions, watch his behavior for a little while but i'm sure it's innocent enough, maybe she's trying to add your fiance to try and get to one of his mates?
Keep calm, he's engaged to you, not her so i'm sure he wouldn't want to do anything to mess it up! My advice is both me and my fiance know each others passwords for everything, mainly because we keep the same password for everything. Knowing each others passwords doesn't mean you have to check up on them because that can cause problems! But it just means you both have nothing to hide and you're open with each other
Wishing you all the best!