Let it go
You have made a fool out of yourself. You've done something you regret. You feel embarrassed because of something you said.
Guess what? People have already forgotten about it.
No one remembers what you did or didn't do as hard as you do. People move on, and so should you. Take lessons and keep going.
People don't scrutinize every little detail and turn it inside out. They don't study the past with a magnifying glass and take notes. You're not the center of their world 24/7.
People don't think what you think they think about you. You're not a mind reader, and even if you were, you still wouldn't be able to make them think differently.
Don't worry about the past. Be grateful about the present.