Should I break up with my boyfriend over this issue?

Anonymous
This is a long story but I’ll try to make the context short. Before we started dating I always saw his ex on his recent searches, he talked about her all the time, and about two weeks before we started dating in January I saw him like a video basically saying he’d always go back to her in the end. We start dating, and he continues to always talk about her and I always see her in his recents, he still likes her photos, etc. He doesn’t talk to her though because she made it very very clear she does not want to be his friend in October, she told him to f*ck off and a bunch of mean stuff. Bottom line this girl made it VERY clear she does not want contact with him, and he also told me she is a c*nt, and she hurt him so badly. In March I finally spoke to him and told him how this bothered me and told him to stop searching her and stop talking about her, his defense was he missed her as a friend. So after this he stops, I never hear a word about her again. They don’t talk for 7 months and I recently found out he reached out to wish her happy birthday which lead to me talking to him and discovering he's “obsessed with the idea of wanting to be her friend” and he still wants her in his life but he “doesn’t have feelings for her” he also still followed her mom and whole family because he wants to see what they post, he also admitted he’s stuck in the past and how their relationship ended. Besides this we are so happy and I love him a lot, but the fact he made no change since March and also continued to stalk her is beyond me, and I don’t know if I should give him another chance to change. He says he’ll do anything, delete her and block her on everything. What do I do?
Should I break up with my boyfriend over this issue?
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