It means that the doctor can save himself.
But with the shinigami eyes, if you got investigated, then you cannot be saved by the doctor (even if you WERE the doctor).
I'm trying to follow the DN universe as closely as I can here, so there really CAN'T be a doctor. however, Naomi Misora was able to avoid getting killed by Kira by using a fake name, Shoto Mako, I think. So I just made up the power that she can hide her true name from Kira in order to avoid getting death noted.
I also figured that Kira would do all his killing by looking up people on the internet, so I reasoned that Naomi Misora could "protect" people by erasing people's public profile or at least giving them fake pseudonyms so Kira wouldn't be able to find someone's real name.
So if someone with Shinigami eyes investigated you, your true name and role would be revealed, thereby leaving you open to getting killed.
