First, do not simply pretend you never had a fight. Why? Because it will come back to haunt you later. Deal with it and move your ass on.
If you are not sorry – do not apologize. If it was truly, yes truly, not your fault – do not apologize. But, if it was indeed your fault and you were being a jerk – Apologize.
Own your mistake and take care of this – Apologize. Apologizing is NOT a sign of weakness. It is a sign of prudence and being smart!
Try to be as specific as possible in your apology. When you apologize, be specific about what you did that hurt your friend. Don’t just blurt out, “I’m sorry, this will never happen again!”
Don’t hold a grudge, and keep bringing this up every time you get into an argument. Move on or this one fight can define your life for your next foreseeable future.
Oh, and it never hurts to get them something nice.