problems with my boyfriends ex!?

[Amanda Papageorgiou]

Hello!I am in a relationship with someone,whose ex girlfriend is still his friend on fb and i think she is stalking on him through fb.She likes everything he posts within some seconds,sometimes she even comments.
the conditions of their breakup (in august of 2013):after living for 4 years under the same roof, he had to leave in order to study abroad, so they broke up.he told me that he didnt love her anymore and that she wanted to have a family with him etc but he never wanted that.He never talks about her, but i think that they had a very boring life together.But when he went back to his country in october for 2 days,they had sex again (he mentioned that she asked for it).
Later he met me and at the end of november we decided to become a couple.one day she asked him to have a video call with him because she wanted to tell him sth.He called her and she said that she finally found someone she really loves and asked him if he is fine with it.he said yes and also that he is with me.now she told him that she broke up with that guy.i dont know how often they chat,he says they dont.On fb everyone can see that we are a couple now,but i never post anything on his wall and we havent posted any photos of just the 2 of us yet.I have to admit that he never talks about her and he hasnt given any reasons to me to be jealous.Its just that im insecure, because permanently we have to live in different countries and we see each other every 3 months.Thanks!Ill choose the best answer.
ive already asked him to unfriend her, but he doesnt wnat to because he says that "she did nothing to him or me". i ve also asked him to give me his password on fb, after a lot of thought he said he wanted to give it to me, but i trust him and i think it would be offensive to him to ask access to his fb account, so i didnt accept it. he wouldnt give it to me if he had something to hide.