Should you fall sick while visiting Kyoto, it will give you much peace of mind to know that your doctor is both reputable and can speak your language! Here is a list of English Speaking Doctors recommended by local residents. This list is not exhaustive of course so please add your own recommendations in the comments:
Kyoto Furitsu Medical University Hospital have English speaking doctors in all department. But you need someone who can help on reception (only Japanese) http://www.h.kpu-m.ac.jp/doc/practice/department/sika.html
Nakano Sensei- who is a regular medical doctor who specialized in gastroenterology but is now a Chinese medicine doctor. I like him because he knows both systems. He’s located on Kitaoji at the southwest corner in the building with the eyeglass store on the first floor. The place is called RokuGoKai.
Kawanaka Sensei at Shugakuin Byoin (Shirakawa Kitayama). Good man, speaks English. I don’t know if the receptionist speaks English.
Dr TANAKA, Kanpo (Chinese traditional) medicine: http://homepage3.nifty.com/rokkaku-tanaka/
Japan Baptist Hospital has some English speaking doctors but apparently the waiting time is long (though in all fairness this will probably be the case in most places). Dr Suzuki in Gynecology, speaks both French & English, and is particularly recommended.
Tel: (075) 781-5191
47 Yamanomoto-machi, Kitashirakawa, Sakyo-ku, Kyoto
More hospitals and doctors are listed here: English Speaking Doctors.