|If you and your manager snapped at each other, you should definitely meet with him to discuss what happened, and make it snappy. At this point, your encounter is an open wound, and if you leave it untreated, it will only become worse.|
Instead of wondering if you should meet with him, the real issue is what to say when you sit down with him. Before doing so, organize your thoughts as to what really occurred, and be prepared to deal with fact and behaviors, and not opinions and assumptions.
Open the discussion by thanking him for taking the time to meet with you. Tell him that you are most interested in working productively with him again. Walk him through the specifics of the incident, and tell him how you honestly feel. For example, if you were embarrassed or humiliated by the public reprimand, tell him. If you feel that your performance did or did not merit some negative feedback, tell him. And, if you feel that you may have overreacted by snapping back at him, tell him that too. The fact is that you both made a mistake. He should not have reprimanded you in public, and you should not have snapped back at him in public.
Let him know that you will do all you can to perform at a level that does not merit any reprimands, while committing to demonstrate more self-control when receiving such feedback. Having made these conciliatory commitments, tell him that you and your fellow employees can be far more productive when feedback is provided in private. Ask him what he thinks, and then be quiet and listen.
If he makes a commitment in return to try to privatize the public feedback, that is a step in the right direction. However, if he cannot manage to take this small step, the fact is that he cannot manage. This does not call for a snap decision on your part, but it does have implications regarding your near-term career plans.