Martial Medic

In order to better learn to heal and treat patients, modern doctors spend years in difficult, specialized study before they are even allowed to begin practicing. Not for nothing is their profession among the most respected in the world. Surely these are the people you most want to turn to if you find yourself afflicted with some serious injury... right?

Nope, in fact, the people with the greatest expertise in fixing injuries are those trained in inflicting them. Particularly if he or she follows the Path of Peace, a martial arts instructor is liable to be far better versed in treating physical trauma than ordinary doctors, and an Old Master will probably turn out to be a medical miracle worker.

This has some basis in fact; traditional kung fu training was often accompanied with instruction in Chinese medicine, and jujitsu masters were often trained bonesetters. Any good martial arts instructor should have some idea of how to respond to injuries likely to crop up during training. However, in fiction, this tends to be exaggerated to the point of martial arts masters offering the best medical care available, along with ridiculous fare like punching people to cure deformities and illnesses and the like.

Contrast Deadly Doctor, an actual doctor who uses their medical knowledge to harm, and Combat Medic, medics who also happen to kick ass.

Anime and Manga

 * Ma Kensei and Akisame Koetsuji of Kenichi: The Mightiest Disciple are masters of Chinese medicine and acupuncture/bonesetting respectively. Together, they have the skills to revive Kenichi from any trauma that occurs to him during training up to and including death. Koetsuji also fixed the boxer Takeda's lamed arm after doctors told him it was irreparable. To say nothing of
 * This is taken even further by the villainess, Kushinada Mikumo, whose Jujitsu style apparently also teaches the secret of eternal youth as she is implied to be around the same age as Furinji Hayato, having fought alongside him in the past, and yet doesn't even look middle-aged. It's also implied that part of this involves some form of dietary restrictions as her disciple, Chikage, is frequently overwhelmed by the novelty of sweet foods that are rather mundane to everyone else.
 * Sanji of One Piece isn't a medic, but his kicking skill is such that he can remodel your face and body with magical plastic surgery. Yes, that's right, Sanji can kick the ugly out of you.
 * And in something of a reverse example, Chopper used his medical knowledge to inflict maximum damage to the giant zombie, Oz.
 * Kenshiro of Fist of the North Star can use Hokuto Shinken to heal as well as to make heads asplode, such as what he did for Lin in the very first episode, who was rendered mute due to the trauma of having her parents murdered right in front of her.
 * Partially justified due to the fact that he's essentially performing a type of acupressure.
 * The medical ninjas in Naruto, particularly Tsunade, use ninjutsu skills for healing, although the significance is limited by the fact that all the important characters are ninjas.
 * In the martial arts manga Kenji, Kenji's grandfather treats some injuries he picked up in a fight with Chinese medicine, noting that he's "still much better than your average doctor."

Comic Books

 * Lady Shiva, a martial artist and assassin in The DCU is also a skilled medic, and has been shown doing everything from reviving a man who had been shot and drowned to relieving neck pain. This is explained as a result of the intensive knowledge of human anatomy that is required for her particular fighting style.

Fan Works

 * In Desperately Seeking Ranma, the most skilled martial artists on Earth develops a ki-based healing ability out of necessity. This ability only becomes more powerful, and usable on other people, after they take formal medical training.

Film
"John Connor: You know what you're doing? Terminator: I have detailed files on human anatomy. Sarah Connor: I'll bet. Makes you a more efficient killer, right? Terminator: Correct."
 * Mr. Miyagi (healing palm) and Mr. Han (Chinese medicine) from the original version and remake of The Karate Kid.
 * Terminator 2. The T-800 is treating Sarah Connor's injuries.

Literature

 * In Terry Pratchett's Pyramids, when the protagonist starts having supernatural symptoms, his friends take him to the doctor and, when he fails to identify the problem, to their school - the Assassins' Guild.

Live-Action TV

 * Played straight in Leverage in The Bank Shot Job - Eliot is the only member of the group whose job regularly involves violence (actually, his job basically is violence) so it stands to reason that he's the one who's going to know what to do with a gunshot wound.

New Media

 * Fraternal twin heroes Yin and Yang, from the Global Guardians PBEM Universe.

Tabletop Games

 * The Path of the Healthy Tiger from Feng Shui is split into two main paths, one focused on healing and the other on counterattacks. A Tiger master is definitely an example of this particular trope.

Video Games

 * The Monks of Etrian Odyssey III are that game's main healers, and their unique "Form Qi" skill makes their healing abilities that much more effective. On the other hand, the Fist Mastery branch of their skill tree exists for a reason.
 * Dr Q. Huaong in Jagged Alliance 2 is also one of the few mercenaries with the martial arts skill.
 * Fallout 2, Fallout Tabletop RPG and Fallout Tactics has "Living Anatomy" perk. It requires some medical skill and gives bonus both to healing and to damage against living creatures.
 * Citan of Xenogears has a bit of this, serving as both the local doctor and Fei's martial arts teacher. He's also one of the best fighters in the game, in and out of gears.

Web Comics

 * Dr. McNinja often uses ninja training to treat unusual afflictions, like a spirit-possessed ear.

Real Life

 * In medieval Europe, the one person in the local lord's employ most likely to know how to treat severe injuries effectively was the local torturer. This makes sense when you understand that his job was to keep very badly injured people alive.