Taking a private fight public moves it to a courtroom with no judge.

Once a disagreement has an audience, the two of you stop talking to each other and start performing for the people watching. Positions harden, because backing down now means backing down in front of everybody, and no one ever arrives to rule on who was right. A solvable problem becomes a reputation contest with no verdict and no end.

If something has already gone public, pull it back into a direct message or a walk. Say the thing you would have said with nobody watching. And if you are tempted to post the screenshot, assume it buys agreement from strangers and costs you the person.