Can I sue the dentist if he left wisdom tooth fragments in my gums?

I am sorry that you are facing this situation. I do not expect you will be able to get a "yes" or "no" answer within the scope of a site like this. An attorney experienced in dental malpractice cases will need to consult with you, review records and, ultimately, obtain an opinion from another dental professional (which is a prerequisite to filing a claim). The analysis will include, among other things, whether this is an ordinary risk of the treatment, whether the problems can be corrected and whether there is a permanent injury (and the extent of it). An experienced attorney like the one who I usually turn to will have both the experience and the contacts for a proper evaluation of the claim.