Calling out a friend in private is respect. Doing it in public is a show.

The same words do different work depending on who is in the room. Said quietly, between the two of you, the point stays the point and they can take it in. Said in front of others, it also asks them to defend their standing, so the one thing they cannot afford to do is agree with you. An audience changes the purpose.

Save anything real for a moment when nobody else is listening, even if that means waiting a few hours. If you have already done it the other way, say so afterwards.