He is right. He was insulted and the other person did not seek to remedy the situation. On top of it, he may feel outnumbered by you and your sister and her boyfriend. So, he only has you to turn to.
Perhaps try to see if your sister's boyfriend can apologize and face the situation man to man. What had happened is in the past. Why ruin the present and the future?
You can not solve this for them. They have to find a way to do it. You could mediate it somehow with your sister's help.
Why can he do the same is a question with many possible answers- male pride, stubbornness, feeling that he is right and you are wrong, not wanting to seem that he is giving in, a level of immaturity, etc.
You are right- you can not live in the past. He is holding the relationship there for his own reasons. Try to let him see that relationships go through many changes in order to grow. No one is perfect including you or him. If he is so inflexible, you would have to wonder how the two of you can make things work out if another situation comes along. Let him know that you have to be able to rely on him and that trust issues are mutual.