First, there were some great posts for gaining forgiveness and reconciliation. I'll leave that alone.
Next though might be a little rougher. You called it, "protective dad mode". I'd like to question that a bit in a moment.
My version was my little girl and her mother (my now ex-wife) with anger issues. 13yrs and I'm shaking a bit just thinking about it. She did hurt her in an out of control moment. That scream that you just know isn't a booboo woke me up and I flew into action from dead sleep already knowing the basics.
Now, here is the difference. I would have been in protective mode but, mom was already out of the room. My first concern was my daughter. I comforted her, calmed her down, checked for injury (yes there was minor injury). If it was another child, I would have waited for her to calm and get a clear story. There was no need. Nothing could justify this. I had nothing more to 'protect' in the moment. I stayed with her for about 30 minutes.
Protection, evaluation, comforting over. Then I went into a different mode. No, I was not violent or loud. The look on mom's face told me what she saw in me with my slow, measured, restrained but clearly enraged direct command that this would never, ever, happen again. She looked scared, and she needed to be. If it ever did, I would be in vengeful dad mode. That's what you were in.
Be honest with youself. Protection, comforting, tending and mending are different than justice, punishment and vengence. All are okay in their time and place but, keep them straight and prioritized.
In the movies the good guys always tend the wounded (if there is time) before moving on. You had time. Your daughter was with you and out of danger.
You said this was the first time. Cut youself some slack. I knew the potential was there so I was mentally prepared and rehearsed. I knew what I might do and what would result. Jail, divorce (which came later anyway), and separation from the little girl I had to protect.
A really wise man once said, "Be angry and sin not." You were caught by surprise. It'll happen again. Think on this and be prepared. I'm sure you're most of the way there already.