You can, but it will not be inexpensive and you will have to prove actual malpractice rather than simple negligence or simple error.
This is usually very difficult and few medical malpractice attorneys will take a contingent fee case and cover the expenses unless you have a substantial claim and it is likely to prevail.
The expenses alone (exams, research, expert testimony, documentation, etc.) can often run into six figures. Do not imagine for one moment that an attorney is willing to front these expenses for you while betting that you will prevail AND collect an award so they can get paid.
Realize that your surgeon's malpractice insurance company's attorneys will present a fierce and thoroughly competent defense. You not only have to prove your case, you will have to overcome their defensive tactics in the process. It is often easier and less time-consuming to push melted butter up a live wildcat's behind with an icepick. (You probably needed some humor.)
Medical mistakes occur frequently. They very rarely rise to a level that results in litigation and, even more rarely, result in a successful outcome for the plaintiff.
You also have to consider your actual damages and the degree to which they were directly caused by your surgeon's alleged error. In this instance, was your ovarian cancer the direct result of incomplete excision or was it coincidental simply because you still had an ovary remaining when it should have been removed?
Compare this to a motorist striking a car parked overtime at a meter. That the car should not have been there does not relieve the motorist of their duty to drive with care any more than your ovary still being present makes your surgeon responsible for you having contracted (unrelated) cancer. If excision was specifically to prevent or correct the cancer you contracted, that might be a different matter, but is still not necessarily litigatable as malpractice.
I am not attempting to discourage you. I am only trying to show you that there is a significant difference between alleging error and proving malpractice. Malpractice is a willful or grossly negligent medical error that significantly deviates from reasonable and customary medical practice AND results in significant and permanent injury to the patient.
Talk with a couple of medical malpractice attorneys. I can imagine that you are probably very distressed. Do not be discouraged because of what I have said, but moderate your likely emotional state with some reason. That you're upset does not convince a court or jury as much as proof. Allegation is not proof. Circumstantial occurrences are not proof.
Your emotional state might be better-served focussing on recovery from your current situation. Malpractice litigation is more likely to upset you further.
Best of luck.