A heartfelt apology includes REMORSE. Remorse comes from true empathy for the pain the other person is feeling because of your actions.
Remorse is different from feeling regret. Regret has to do with wishing you hadn’t taken a particular action or said something. You may regret an action because it hurt someone else, but you may also regret it because it hurt you, it cost you something emotionally or financially, or led to a punishment or undesirable result. Remorse is only about the other person.
After you’ve said your apology, just let it be. Don’t backtrack, don’t add caveats or excuses, don’t add conditions. This is not about you and your ego, this apology is about the other person.
WHAT NOT TO SAY
-”Yes I was wrong *but* you have to admit I’m not the only one who was wrong here.”
-“Yes I was wrong *but* in general you have to admit my point still stands.”
-“Yes I was wrong *but* it was wrong of you to make a big deal out of it.”