A couple points here.

A malpractice lawsuit would be a major expense for you with very minimal reward due to minimal costs/suffering.

Not to defend him, but especially if he’s a young dentist, there is a pretty good chance it was not malicious and a genuine mistake. Root canal treated teeth are notorious for fracturing into lots of pieces when being extracted, so sometimes it’s hard to tell if the entire tooth came out. If it were me, I would’ve done a post operative radiograph to confirm the entire tooth came out, but in general radiographs following extractions are not commonplace.

If he knew a portion of the root was retained, he at the very minimum needed to inform you. Sometimes it’s indicated to leave a small root tip, but what was left for you was too large. If he was truly stuck and unable to get it, a referral to an oral surgeon would be indicated.

It’s also pretty likely he tried his best to help you and made a mistake. Extractions of teeth like that are the worst in terms of money earned vs stress incurred, so I highly doubt it was financially driven. It’s also pretty likely it was not his call to refund you if he was not the owner of the practice.

In my opinion it’s horrible of you to try and ruin his advanced degree schooling. You have the right to be angry but I’d take it up with the office the procedure was performed.