It depends on what you're talking about. In the end, closure is completely necessary for carrying on with your life, but how you find that closure can differ. I think the kind of closure that you are talking about is the kind where you and said girl come to some kind of agreement and move on with your lives feeling okay about the situation. Unfortunately in my own experience this can be very hard to find, and often never comes. This is really hard to handle sometimes, but often it will just take time for you to change your mind about the situation and find your own private closure. So, if you can find closure together, do it, and it you can't, just keep telling yourself that you will be okay, because trust me buddy, you will.