In most jurisdictions, a medical doctor or dentist does have the right to veto settlements. In my opinion, this is what has caused the rising cost of malpractice insurance. If you have a lawyer, it is best to ask him or her. If you do not have a lawyer for this case, I would hire one immediately.
I concur to Mr. Adams that you should consult with your attorney. But, I am confident he or she will tell you the following:
Under the court orders governing mediation, parties have to mediate in "good faith" and attend with the decision makers armed with settlement authority. Beyond that, there is very little required structure beyond that dictated by the individual mediator. Parties often walk out of mediation if it is apparent the case has no chance of settling. It cuts off the mediator's meter as it were, since the parties split that bill. Unsure what you mean by a "formal 'closing' statement" but, again, so long as a good faith effort is made and appropriate representatives attend, pretty much anything goes. Good luck to you.
Mediation is just one step in the course of litigation, and unfortunately some dentists feel put-out by having to participate at all in the process when they are sued. So, the dentist in your case was probably told he had to made an appearance at mediation to satisfy a court order. So he put in the least amount of effort possible and walked out, which might have been in bad faith, but not to such an extent that the Judge would likely deny him a defense at trial.
Good luck with him. In my experience, dentist could destroy a patient's mouth but still deny any wrongdoing. Thankfully his insurance carrier might be more reasonable.
The parties' right to self-determination is paramount. If both parties attended in good faith with full settlement authority, such an impasse is permissible, though mediators should generally try to prevent such an early termination. As for consent regarding settlement by the insurer, that depends on the contract terms.