Can a dentist charge a patient after they have already accepted a co pay and later finding that the insurance will not cover the procedure?

It depends on what the agreement was between the patient and the dentist prior to the procedure.

When you first came to the office, you filled out and signed a lot of paperwork. If you signed an agreement that you would be financially responsible for any fees not paid by the insurance, then you are on the hook for the balance.

"I thought someone else (the insurance company) was going to pay for my dental work," will not stand up in court, and you run the risk of hurting your credit if the dentist obtains a judgment against you.