And that's how you'll choose your doctor anyway - most probably based on the word-of-mouth recommendation from his/her past patients. Since there are the Eric Rudolfs, the state has legitimate interest in not making their job any easier.
What would you think of a physician rating service? Rates allergists, cardiologists, internists, abortionists...
Shouldn't women have the right to go to a good abortionist?
Anyone who wants to murder an abortionist can easily find one in the phone book. Do you think that abortionists' offices should be unlisted?