How can I save my relationship?

Is commenting on a girl's picture that she looks cute a reason to break up with someone? Absolutely not. I don't see what the big deal is, if your girlfriend really liked you she wouldn't break up with you over something like that. When people like someone (at least when I do) they're ready to forgive that person a lot of terrible things just because they don't want to lose them. Seems like she wasn't afraid of losing you. I'm sorry buddy, I just don't think there's anything to really save here, just give up, your relationship is over, she wasn't the right one for you. You shouldn't be treated like that anyway. If she likes you maybe she'll come around eventually, but personally I don't think you did anything wrong therefore there's really nothing you can/should do to "fix" it.

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You can't she does not trust you or she was looking for an excuse accept it's over and move on

If what you say is true, she is a little over sensitive. YOU should be a little more careful if you wish to avoid this kind of grief again. If you are unable to fix it, learn your lesson and be GLAD to be rid of her. Look for someone a little more stable next time.

Unfortunately for guys you say "cute pic" but girls see "you're so hot". Just try to explain to her that you weren't flirting but just trying to be nice. If she accepts that fine, if not then you may not want to be with her anyway.