My first thought would be that he fancies her. But I would definitely ask him about it
I wouldn’t be concerned if it was once or maybe twice i saw it as i do it quite often, i will randomly think of someone from my past and have a little look online what they’re up to. But if he is consistently looking her up and also like you say falling asleep looking at her page i would definitely be having a conversation with him? i wouldn’t go in there attacking or questioning, i would just mention that like u moved his phone n saw she was on there, how’s she doing etc. By his reaction you’re more than likely going to be able to tell if it’s something he’s feeling guilty about so maybe something bad or if he genuinely doesn’t seem bothered that you’ve mentioned it, he may just be having an innocent little nosy at what she’s up to xx
Is it part of grieving. Does he feel guilty for blaming her? Does he wonder if she's doing ok because maybe he's still hurting? I'd guess their grief may be similar in some ways being so close to the man who passed away. Hopefully it's something along those lines and nothing else. I'd speak to him? Check in on him? He may be missing his friend? Honesty is always best. If it bothers you talk to him about that.
Omg im so nosy I literally look at the most random people, including exes, and I would be mortified if my husband thought I was doing it because it meant something. It absolutely does not mean anything to me - I am literally just bored and nosy.
if his friend died, it's probably just a grief thing. they are the 2 closed to the friend.
Have you spoken to him about it? That's the first thing I'd be doing?