I don't know your particular situation. I don't know how things ended between the two of you. Think about how it ended, and then put yourself in her shoes. If you were her, would you WANT to hear from you? Actually reading your message, you make it pretty clear that telling her how you feel is out of the question. What you are wanting, obviously, is a sense of closure, and a chance to express your remorse over a past wrongdoing. This can be very heavy to carry around, I know, I've been doing it for decades now. If apologizing to her directly is out of the question, then the best thing you can do is live your life as a good and decent person who treats people well. Make amends to her by being kinder and gentler to those who are part of your present. Write her a letter (but don't mail it) just expressing your thoughts. Write it as though you are going to mail it. And then when you find yourself thinking of her, go read it. Also keep in mind that no matter what happened you are not responsible for her well being today. Don't let yourself be weighed down by guilt over something you simply cannot change.
I had a situation very similar to yours. I definitely, definitely know how you feel. I have those recurring thoughts about two ex's actually, but one that was definitely more prevalent in my mind. I was lucky in that I knew his wife, and was able to pass along a "I hope you guys are very happy and blessed" type of message that I had been thinking about for years. It was well received, and it's her decision whether or not to pass those well wishes onto my ex. I don't need to know either way, but my heart sure feels a lot lighter knowing that I have been forgiven.