A malpractice lawsuit is a Civil Lawsuit.
When you sue someone for malpractice you go to a Civil court proceeding and sue them for the malpractice.
For a malpractice lawsuit the alleged victim of the malpractice lawsuit brings the lawsuit for the purpose of remedying perceived damages sustained via monetary compensation.
There is no state involvement in the lawsuit unlike a criminal lawsuit.
Malpractice itself is not a crime so it would not be a criminal lawsuit or criminal court proceeding.
You would need to find a lawyer that deals in Civil or malpractice lawsuits.
It can be hard to sue for malpractice but it can be done if you have enough evidence.