Just got into a big fight and I was wrong. What should I do going forward?

My girlfriend and I just got out of the biggest fight in our relationship and I was completely at fault here. I disrespected her and her friend by checking her out, and now she questions my faith and love for her. There are other things that I had done, but I don't want to go into any more detail as I feel embarrassed and guilty. The worst part of all of this is that I pulled every "do not do in a fight" card. I was so sure that I did nothing wrong, I was defensive, hypocritical by bringing up past arguments, and never bothered to understand her feelings. I felt that my feelings were hurt by some of the things she said and I put my own feelings before hers. If I truly cared, I would have tossed my feelings aside and really tried to understand what she was going through.

I admitted to my mistakes, pointed out everything I had done wrong, and apologized. We are still talking, but she is still very mad and annoyed at me right now. And she has every right to. She hasn't given any big indication of a breakup yet, but I'm preparing myself in case that happens. Regardless of all this, I want to move forward with all good intentions for this relationship. I truly and genuinely love her very much and I want to show her that I still care for her. But more than anything else, I want to show her that I respect her.

My thought is that I shouldn't rush things and keep things simple and start back off as if we were friends. Have a quick and simple coffee date, go hiking, maybe go to a local event with food trucks. That is, of course, after I've given her the space and time that she feels she needs. Ultimately, I really want us to be back together, but I also don't want to force anything on her.

What is everyone's thought?
If you've been in a situation like this before, what did you do to make things better?
Is it fair for me to ask when we can follow up on our relationship?
What would you recommend as a good step forward?
Just got into a big fight and I was wrong. What should I do going forward?
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