I noticed that while Arzt has a female equivalent, Ärztin, the female equivalent of Doktor seems conspicuously absent. The DWDS usage database does list examples of Doktorin in "die Zeit" going all the way back to 1947, but it's use continues to be rare and sporadic. It seems to be more popular in the spoken language though, judging by its appearances in subtitles. While Doktorin does have an entry in German and English Wiktionary, it's missing from DWDS. I won't ask people to speculate on why, but I was wondering if Doktorin shows signs of catching on, dying out or is it just treading water. Would it seem odd or normal to call a female Ph.D. der Doktor and if you did would she give you a nasty look? Is there a gender neutral word that's replacing both terms?
PS. Based on the answers I refined my searches to get a better understanding of what's being used. First, I restricted the search to 1998 and later to remove any mid-twentieth century style sexism. Then to eliminate the form of address vs. description issue I searched separately for Herr/Frau Dr./Doktor and Doktor/Doktorin der; almost everything you can get a doctorate in is feminine (somewhat ironically), so der should cover most descriptions of the form "Doctor of X". With the Herr vs. Frau searches the ratios were about the same as the ratios for Males vs. Females getting Ph.D.'s; see David Vogt's comment under one of the answers. For the -or der vs. -orin der results the ratio is much higher in favor of -or. I originally had the impression that Doktorin was more common in the subtitle corpus, but now I think that's incorrect.
For what it's worth, I did find a specific example where a female Ph.D. is called Doktor. It's from the series "Terra Nova" which I recently watched so I remembered who it was about. The quote is about Elisabeth Shannon, mother of three and one of the main characters, and the original English, from a fansite, is
... doctor of science in immunology, bacteriology and, couple other "ologies" I can't even pronounce.
The subtitles render this as
Doktor der Immunologie, Bakteriologie und... ein paar anderen Ologien, die ich nicht mal aussprechen kann.
Perhaps the translator didn't know the context and assumed male, but it seems unlikely.