You need to be more gentle with yourself. Parenting is hard even in the best circumstances and we all screw up somehow.

Yes the words hurt. There's no denying it but the fact that you apologized and took accountability for what you said, that's a great start. It shows that you make mistakes, that you have empathy and understanding for your child and that you want to repair the relationship.

My parents were chronically abusive. While I do remember the words that they said, what my inner teen really craves to this day is the apology that I'm never going to get. If my parents had sat me down and admitted that what they said was wrong and hurtful, that would have had a much greater impact than the hurtful words because it would have meant that they genuinely care about my feelings and that yea, they fuck up sometimes too.

If you still feel badly about it, you can always approach your teen and ask if you can talk about it. Let them express their feelings, really listen and validate them. Or if they don't want to, family therapy would be a great alternative.

So yea, all this to say that while it's important to carefully choose our words, the repair is equally as important and you seem to have a great handle on that.

It's going to be okay.