Points to consider:
1. European martial arts traditions were superseded by the use of firearms (certainly in the armies).
2. From the sixteenth century, Europeans experienced almost continuous warfare often using conscript armies (or at least large numbers of conscripts). An army cannot waste its time teaching a fighting system that takes ten years to master to conscripts that need to learn to fight in eight weeks. European armies discarded all but the simplest fighting skills - plus the gun of course.
3. In the East Asia (China, Korea, Japan) the same period of history was one of relative political stability (at least when compared with Europe) and limited technological advancement in weaponry. As such, mediaeval traditions of combat continued much later. Hence the emphasis of armed and unarmed combat arts, like Te, forms of Tae Kwon Do, Kung Fu etc. The lack of rifled bore firearms meant that such skills remained relevant to the battle field and the maintenance of professional warrior classes or groups (samurai, shaolin monks) meant that arts that took ten years to master were still effective.
4. Numerous unarmed forms of combat exist from Europe, most famously savate - french kickboxing - and capoeira - which was developed by slaves in Brazil.
5. Many cultures have traditional forms of exercise designed to keep the men of the culture in shape for warfare. Southern african nations had running; Greeks developed the Olympiad with games originally based on soldier's training (modern athletics derives from this tradition); there's a form of exercise from the Ottoman empire still practiced today in Turkey (I've forgotten the name); the Highland games in Scotland; in the case of East Asian countries and cultures, martial arts are this form of traditional exercise.
That's about as much as I know from the history. Someone's previous post about GI's bringing karate etc back after WWII/Korea is also an important point. East Asian martial arts are not the only ones, just the most well known.