Boyfriend searching his past love on FB

Hi soompiers!
There's something that's been occupying my mind recently. I've been going out with my boyfriend for nearly 2 years. I love him so much and I believe the love is reciprocated. We've been through a lot of horrible things because of his past. We fought continuously and thought about breaking up with each other countless times. The problem was, last year, while we were still going out, he preferred to see this one girl he had a crush on for years, and never saw me over the summer. He never thought about meeting me in the summer and decided to go out with this girl one-on-one in the city. I know he has liked her for the longest time. She's pretty, smart, and ridiculously nice... to the point where even when she had a boyfriend, she would get all touchy touchy and flirt with other guys around her. Of course, her relationship broke apart and it seems like her bf gave up on her because of that. I don't know her personally, but by the way she talks to my boyfriend... makes it really sketchy. She's clingy and likes to act cute to guys all the time, sending them hearts, winks and calling other guys "bae."I'm my boyfriend's first. He never had a girlfriend before. He was going to ask her out years ago, but never did because of distance issues (and college happened). My boyfriend is the type of guy that falls for girls that are extremely nice to him. He also gets extremely attached to those type of people. But because of our continuous fighting, he hasn't been talking to her. I don't think he contacts her anymore (but that's because I told him how extremely uncomfortable it is for me). So technically, I stopped him from contacting her. He didn't delete her contacts through his initial intentions.I've told him calmly that I love him so much and I appreciate the time we spent together. But I also delivered my thoughts on how it's uncomfortable for me to see him search her up on FB.To be frank, I know he's curious about her. I don't mind that. I really don't. If he's just searching her up through curiosity, then that's completely fine! But what I want to know are: if he still likes her, and what he still thinks about her. He's told me before that he didn't like her anymore, but it's hard for me to believe that because every time we bring up this subject, he gets defenseful and angry. And whenever he's angry, he tends to lie about things. So I am not sure myself if he is telling me the truth when he says he doesn't like her anymore. Even when I asked him if he missed her, he didn't say anything and said he wanted to stop talking about this.Even when I approach the situation calmly and understandingly, he just raises his temper and says he doesn't want to talk. He won't tell me anything. I hope he can understand why I'm asking these things... I just wish he can have a mature conversation with me, instead of walking away and doing other things and avoiding me. Most of the time, it irritates me how immature he can get. I just want to know if he misses her, if he wants to see her, if he wants to be with her instead (because of all the fighting we go through). I have asked these to him, yet he just ignores me because he assumes that we will end up fighting again, and he doesn't want that.
So I'm here asking you guys on what you think. Hopefully I can get some advice on this situation. Fellow soompiers with prior experience with this situation: please let me know how you guys overcame this, and what you did!!
Thank you!!!!

His defensiveness is a sure sign he doesn't want to argue and knows that the argument will happen because he has impure reasons. If you're willing to argue, usually it means you're convinced you're correct about the situation, but since he is being defensive, he knows it's not the best thing for you, but he's not willing to give it up, and he's trying to avoid the argument because he "doesn't want to hurt you". If that thought ever comes up in his mind, it's because he's pretty sure it probably would indeed hurt you. that's why people hide things, and then get defensive when discovered. So he's not 100% over her, and obviously that's bad for you two- he should really consider you and the relationship you two share more. I know you don't want to leave, you love him after all. but here's the thing, whenever there is a loving relationship and one person does something bad, usually the other can forgive him/her because it's not worth how much love they share. but if they're consistent with this, what happens over time is that it keep chipping away at the existing love, the love decreases, the arguments and reasons not to be together increase. When they reach a compensation point, then break up is the next step. so for you, you should stick it out and stay as long as you want. keep giving him chances to redeem himself, until the compensation point- that's the point where not only can you not take it anymore, but you also don't love him enough to forgive him and continue on. If he can fix himself in the interim, great, if he can't- you will know break up is the only solution. by that time, you will be ready to break up, you will want to break up. and you will break up. it is these cases where the perpetrator finally learns how much s/he has been taking for granted the relationship and finally realizes how much s/he loves his/her S/O, but has not shown it to them. they will usually try to crawl back, and then it's up to you to make the next move. But I don't recommend taking them back easily, because if they haven't changed, then what have you two learned from the break up? that you guys still miss each other or love each other?? that's a given and not enough to take him back. the lesson that should be learned is that he has to stop looking at that other girl or do things that affect the relationship negatively- which btw, usually doesn't happen because people can't change their personalities or feelings easily...

Insitnctually thinking it may be time to begin getting ready to move on. Not even trying to be negative. Just being real.

@bubbletealove
I would like to clarify a few points:
1) Your BF has stopped all contact with her, but you know he is still looking at her FB?2) Out of the blue, you will bring up this topic every so often because you are unsure how your BF is feeling towards her?3) other then this girl, you and your bf have no other major concerns in your relationship?

Based on my assumptions from what I have read, here are a few points I would like to make: 

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bubbletealove said: Hi soompiers!
She's clingy and likes to act cute to guys all the time, sending them hearts, winks and calling other guys "bae."