Regardless of how close you maybe with someone, the time will inevitably come when one of you will hurt the other person’s feelings. It is a given. You probably didn’t mean to hurt their feelings. Perhaps you don’t really care that you hurt their feelings; but chances are, if you are close enough you do care.
The worst thing you can do is invalidate their feelings by pretending it did not happen or that they have no reason to be upset. Some people are more sensitive than others. Perhaps they are going through some personal hardships or have had past experiences where others have treated them badly and are over sensitive. Whether or not you believe you are in the wrong, the fact still remains that their feelings are hurt.
The easiest thing to do, although it is hard for some people, is just to apologize. You do not have to necessarily admit you did anything wrong if you feel you did not. Just tell them you are sorry if you did something to hurt them. However, if you did something wrong and you know it, apologize for whatever it was you did. You are not a pansy or weak for apologizing. Everyone makes mistakes but it takes a big person to admit them.
Sometimes, people just need some alone time. If you apologize, and they do not accept your apology, you have done all you can do. Leave them alone and give them time to work it out on their own. If they value you, they will seek you out.
Just remember, this is a two-way street. You should never be the person who is always giving in or always apologizing. It takes a little life experience and being a good judge of character to know just who those people are. If the other person treasures you as much as you treasure them, they will also make the effort to work things out.