You are trying to defend yourself against something that is basically indefensible, with the kicker being that you didn't do it. It's natural to want to "defend" yourself, but you are making things worse by doing this, not better. Confrontation will only make her dig her heels in.
To top it off, your friend sounds like she has some problems with people and relationships in general.
You can't tear yourself up over this. She has something going on in her life, and for whatever reason, she is taking it out on you. You can put the ball back in her court.
Send her this note:
Dear _____ ,
I care for you and cherish your friendship dearly.
During the past 15 years, you have had ample opportunity to observe what kind of a person I am. Over the years, I have noticed that you are very guarded in certain areas. I have consistently respected your boundaries. I am both saddened and deeply hurt that you believe I would have violated your privacy. I would have hoped that after all these years that you would instinctively know that this is not something that I would do. Unfortunately, this doesn't appear to be the case, and I am truly puzzled as to why.
I love you and value you as a friend, and it concerns me that something has happened to make you so distraught as to throw aside everything that you know about my character and believe the worst.
I'm here if you want to talk. If not, please know that I will always wish you the very best.
Love,
Bubbles