Even the best of friends will have differences. Quarrels are also a way to understand each other. As for fighting, it is very normal between boys. No fight, no acquaintance. Fighting with good friends often makes things better after reconciliation. Here are some suggestions for you:
First, call another good friend and have a meal together. There is no need to explain deliberately. Boys are like this. They will naturally understand each other and have a bigger mind.
Second, talk to him alone. Obviously, you are good friends. What is there to be embarrassed about? I think he will definitely regret it. Give each other some space, and it will be easy to reconcile.
Third, give him a gift. You can give him any gift he likes. But it is recommended that you give him a creative gift, a real-life doll of the two of you, made into the style of good brothers. The realistic and cute appearance will make your friends like it very much.