I will be using the phrase "kung fu" to mean Chinese martial art(s). The phrase does not actually mean "martial arts", but rather something like "something achieved by hard work".
The Mandarin phrase that actually means "martial art" is "wushu". However, the use of "wushu" has been complicated in modern times by the much more popular use (at least in the west) of the phrase "kung fu" and the fact that there is a martial art-sport that was developed by the Chinese government called Wushu.
The most widely spread story about the origins of kung fu involves a Buddhist priest/monk named Bodhidharma traveling from India to China in the 5th or 6th century to translate Buddhist scripture.
The story goes that Bodhidharma came to Shaolin Temple, but was barred from entry because he was thought to be barbaric. Bodhidharma then hiked up Mt. Song and meditated in a cave for nine years. When he came back down, he entered the temple and was disappointed to find that the monks would frequently fall asleep while meditating. He taught the monks some exercises to improve their physical fitness, which he deemed necessary for spiritual fitness.
That, supposedly, is the origin of kung fu. Highly dubious, of course.
More realistically, there have always been fighting styles in China (as in all places). Many retired soldiers would become monks in their later years, bringing to the temples various martial techniques that would come in handy when monks might have to defend themselves from bandits. What Bodhidharma really did was bring some yoga exercises, and probably some Indonesian fighting techniques that now make up Pencak Silat. Bodhidharma did not create kung fu, but simply contributed to China's martial traditions, and helped institutionalize the style of fighting practiced at the Shaolin Temple.
(Arguably, Bodhidharma's more important contribution was that he brought with him the teachings of a particular sect of Buddhism that came to be known in China as Ch'an, which would eventually be known in Japan [and all over the world] as Zen.)
There are a LOT of different martial art styles that orignated in China, and I plan to write a little bit about a few of them. (These may include Wing Chun, Baguazhang, Shaolin, Tai Chi, Qi Gong, Praying Mantis, Xingyi, Sanshou, Chin Na, Hung Gar, Mok Gar, Choy Li Fut, Five Ancestors Fist, Tiger-Crane, Wudang, Five Animals Style, and/or Wushu.)
Would Batman learn kung fu? Of course. He wouldn't really know too much in the way of martial arts if he didn't have some sort of training in at least one Chinese style.