Legally, you most likely can: leaving glass in the wound was most likely negligent, or unreasonably careless, since a careful doctor would have checked and made sure all the glass was gone. Medical negligence is malpractice, and malpractice makes a doctor liable, or financially responsible, for any later or additional medical care required due to the malpractice.However, it is questionable whether it is worthwhile suing the doctor--the only way to get the money, if he won't voluntarily pay for the second procedure. If you sue for malpractice in a case like this, all you can get (assuming that there is no lasting significant injury to your finger, as we hope) is the out-of-pocket, or not paid by insurance, additional medical costs. But to bring a malpractice case, you *must* have a medical expert, like another doctor, testify as to the appropriate standard of medical care, what the malpractice doctor should have done, and how what he did was malpractice. Medical experts can be very expensive--easily over a $1,000, possibly several thousand. It is very likely that unless you have unexpectedly large medical costs, that you would spend more on the lawsuit than you would get back (you have to pay your own expert costs; you can't get it back from the other side).