You can end up hurting someone without ever meaning to bully them.
It was a joke, everyone was laughing so you kept going, you couldn't say no. The reasons vary; how it landed is separate.
Sooner Is Better
Apologise as soon as you realise. The longer you leave it, the harder it gets, and only the fact remains for them.
Don't lead with reasons. "I made you feel bad back there. I'm sorry" is enough. Starting with excuses doesn't land. It also goes better one to one than in front of others.
What Follows Decides It
Apologising once means nothing if you carry on the same. But if your behaviour changes, how they see you can change too.
They may not forgive you. Stopping still matters: what counts for them is that it doesn't continue.
If something comes to mind, apologise in one sentence without giving reasons.
At Digital Kodomo BASE, we value being able to stop once you notice.
Check How should you apologise?