First of all - no where does Bubblejunkie indicate whether his ex-friend was romantically involved with him. We also don't know what caused the falling out. We can all assume different things, but from what I gather from his post is that something went down and got ugly and now his friend wants nothing to do with him, but he wants forgiveness and for things to get better one day and/or things to go about as if nothing happened. Well - forgiveness (IMO) is more than likely dependent on what was done to ruin the friendship to begin with. And even if he does get forgiveness, it does not necessarily mean that they will be friends again, and he has to live with that.
I have several ex's that I will NEVER forgive as long as I live and are dead to me now. And when they tried contacting me at all hours, tried to get back with me and made promises that more often than not get broken because they always seemed to fall into the same routine, it just pushed me away further, and put that final nail in the coffin that closed the chapter on our relationship / friendship etc... But I have been on your end as well, Bubblejunkie, and found that if I kept contacting them, I got pushed away even further, and lost any hope of ever being friends with that person ever again, and I think that hurts even more than the initial fight/argument. Ego and Pride has no place in wanting forgiveness for reasons other than just that. Do you have alterior motives behind wanting forgiveness for the sake of genuinely being there for her in a time of crisis, are you being selfish because you miss the times you had and want that back more than forgiveness? Not being harsh but these are questions you have to ask yourself. We don't know much about your friend, but you have to also consider that maybe she needs to be alone and face these issues she might have on her own. Maybe this is vital for her own growth, and strength of character...
My suggestion is to let her be. If forgiveness is possible, the only way you will get it is when she is good and ready. And if you never become friends again, chalk it up to a lesson learned, and grow from it. Learn what you should / shouldn't do when you are faced with a new friendship. What worked in the past? What didn't? If you learned something from this experience, then you have not failed.
One saying that always helped me is: "People come into your life for a reason, a season or a lifetime". Maybe you can figure out which one she is/was and find peace with this situation.
Take Care!