Well, be open, explain why you did it (thought it would be funny etc), and overall, talk about it. If you let it stay unresolved, the problem might act as a rift in your relationship.
I insensitively but unintentionally teased a best friend about having too much to drink the day before. She did not appreciate it. I felt so bad I cried and apologized repeatedly. I still feel terrible. What to do?
This is called a life lesson.
You've apologized to your friend, now let it go. When someone goes on and on about how bad they feel, it makes the situation even worse.
This is your conscience speaking which is good. Some people don't have a conscience so when something bad happens, they don't even flinch. So do everyone a favor and drop it. I believe that the next time you feel like spouting off something, that you're going to think before you say something, which means that you've learned a good lesson from what has happened.