You should know her deepest fears and why she feels that. She fears that you leave her because she has a trauma of her parents who left her behind.
So you should say something which has something to do with the things standing above.
"FEAR", "NOT BEING THERE FOR HER" and "LEAVING" are kind of trance words. So if you want to say something to her, say something the opposite of this.
It's good that you're appologizing for it. You have to give her a reason for your behaviour. She will be more understanding as a result.
I'm not very experienced on this part, because I never had the same situation. But I think you should do the following if you love/like her and want to continue the relationship.
I wouldn't tell her right away that you're not going to leave her. Instead, take her out somewhere, on a date. Act realy spontaneous, happy and energetic. Do fun stuff, go to the fair, go shopping together or go to a nice place where you can eat. Do a lot of fun and exciting and fun stuff together. That way, she will be relieved and thinking "he would not dump me if he is taking me out on this very funny date and act like he is realy loving me". But still, because of her fears, she will be still doubting. She might be concerned that a few days later, the fun will be over again and that you will be acting like you did before, and thus, she will be affraid that you might dump her again.
So then go to a romantic or intimate place somewhere outside, or maybe to "your secret little spot" where you used to meet or date a lot in the past, and then tell her right away what you wanted to say to her:
"I'm sorry about the past period. I was having a rough time and I tought mostly of myself and not enough about you and how you feel. I know I made a hard time for you, but remember: I love you. I realy do. And therefor, I will certainly not leave you. I am there for you"
It might sound a bit cheesy maybe. But I think this is the best thing you can do. If she realy fears you're going to leave her, she should be realy into you. Telling her you love her will do miracles (especialy if you haven't told her before). She must realy feel that you like/love her and that you will not leave her. She will feel secure again as a result. She might feel so happy after you said that that you might make out passionatley and have sex later on

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I would not give her the power tough. That would make her dominant, and that she should decide when to break up or not. If the relationship turns out bad in the fuuture and you want break up, you will be a jerk because you told her that she could decide, that it was up to her. Always keep that decision and the power, because you know how you can use it and if you give it away, you don't know how she might use it.
Besides, it has to be said that there should always be a slight fear that you might leave her. That will stronger the relationship and bring more excitement in it. But keep it under control. You can use this power to punish and reward her if you wish: if she does something good, make her feel that you want to stay with her for a long time, if she does something you don't like or act bad, make her feel indirectly that you have the possibility of leaving her. This is called "willingness to walk away". it's the same principle you use during pickup in a bar.
I have to say that, if you apologize for your behavior in the past, that you also tell her that you will not do it any more. Later, you should not behave like that again for the period/ next (few) month(s) that are about to come, if you want something serious.
Cheers!,
Shyler