Obviously the best scenario in such a situation is to move on with your life, remain in good terms with your ex and even be friends. Deleting her from facebook shows you can't handle the second part yet.
That's nothing to be ashamed of. If you have just broken up from a very close relationship, chances are you are not secure enough to handle seeing new pictures of her with other guys on facebook. You do what you have to do. Cut her out of your life, move on, start getting on your feet again. Once you rediscover your passion in life, appreciate your friends more and even find a new girl, you will be secure enough to face her without feeling bad or falling back into afcdom. Then you can add her back on facebook or even be her friend, if that is what you want. Chances are you will discover that she is not worth it. Personally I would like to remain on good terms, even though we won't be close knowing our differences.