When a patient takes a doctor to court and sues the doctor for malpractice the malpractice suit does become public record where someone can research and pull up on the public record whether the doctor had a malpractice suit or not.
However most cases of malpractice lawsuits are automatically thrown out by the judge because sometimes a patient will try to sue a doctor for malpractice even though no malpractice actually occurred to them.
So just because a doctor has been sued for malpractice doesn't mean the doctor ever did any malpractice nor does it mean that the doctor is a bad doctor.
There are likely thousands of doctors and surgeons working in hospitals and doctors offices today that has had at least one malpractice lawsuit on them but they still are good doctors and surgeons.