Apologizing to your little brother when you’re wrong is real character. Nobody’s making you.

Apologies you are forced into are just compliance, and everyone involved knows it. The ones that count have no enforcement behind them: nobody is standing over you, he cannot make you, and you could wait a day and watch it fade. Going anyway, when letting it pass would cost nothing, is the version that actually costs something.

Go and say the specific thing you got wrong, without the sentence that explains why he provoked it. Keep it short. Then do something ordinary with him afterwards, because that is the part that tells him it is over.