Europe and the Middle East has lots and lots of different types of folk wrestling. On top of that, there is pankrase, the various military combatives from fencing treatises which include grappling, etc. The only reason its not as popular is because 70s Hollywood.
You might find some answers to your question here: http://www.reddit.com/r/AskHistorians/comments/1pdf01/why_are_handtohand_martial_arts_so_much_more/
EDIT: I am embarrased that this top comment actually provides little to no information.
They weren't, it's just that the categorization and longevity of the Asian arts has been better preserved and is better known. It's important to understand that unarmed fighting techniques are basically a "last resort" in military techniques. It means you've arrived at a position where you are deprived of all your weapons, all your comrades with weapons and a means of escape (like a horse).
What we think of today as "empty hand" martial arts are really just a part of an overall armed training program. For example, a Samurai would have trained in pole weapons, bow and arrow, staffs, clubs, chains, horse riding and later matchlock and cannons, and then down at the far end of that training, grappling and wrestling and some kind of unarmed fighting system. http://www.thehaca.com/essays/knightvs.htm#.VChIJPldU68
A western professional soldier would have had similar training in a similar number of weapon classes and techniques. In addition they may have extensive training in combat "games" like jousting. For example, a Knight Templar carried: a sword, shield, a dagger, a "bread" knife" and a small knife, a "Turkish" Mace, a lance, and many carried crossbows or other archery weapons. They would have also been expert horsemen and trained in tactics and strategy. http://www.knight-test.aspery.com.au/armour/knight-templar.php
People point to the vast number of styles of Kung Fu, but most of those were some kind of local or family style - with more similarities in local arts than differences. For example, Kung Fu is today broadly classified into Northern and Southern styles, and each broad set looks and functions largely self-similarly within a broad category. You might say there's really only a few Chinese Arts, and hundreds of variations of each. https://martialarts.stackexchange.com/questions/1366/what-is-the-cause-for-northern-and-southern-chinese-kung-fu-differences
It's a bit like the distinction between languages and dialect, it depends on who's doing the classifying. There's hundreds of "dialects" of Chinese, while there's dozens of "languages" of Romance European (each with dozens of dialects). But linguistically there's no difference, the assignment of language and dialect is arbitrary. Two people from different parts of China might speak what we call different "dialects" but be completely unintelligible to each other. The distinction between fighting arts is basically the same. All 12th century western knights didn't learn the same fighting arts, but today at a modern Renaissance festival we don't bother to make any distinction between an English knight from Wareham vs. a French knight from Mayenne. It's just sword swinging by guys in armor.
The major arts used by the nobility or the military were definitely armed and many modern Chinese Martial Arts spend most of their training time with weapons of some kind. For example modern Wushu trains in short staff, long staff, 9 sectional whip, sabre, broad sword, straight sword, spear, rope dart, halberd, hook sword, trident, mace, hammer and more. To be fair, modern Wushu is also an aggregation of many different family styles, but each of the original styles would have probably trained in at least a few of these traditional weapons. http://www.hcs.harvard.edu/~hwushu/?page_id=5
Going back to Western arts, a knight in training would have trained in his local style as well. The weapons might have been a little different from area to area. A broad sword in one area might have been slightly shorter than in the neighboring town, trading length for speed for example and all technical differences that come with that change.
Are they different styles? In a Chinese Martial Arts context, they probably would be considered different. From a Western Martial Arts context we just call it "broad sword as taught by the arms master at xxxx". But practically they're the same thing. Unfortunately, not a lot remains of those styles. We have some later manuals from the 14-15th century onward. http://www.thearma.org/manuals.htm#.VChLIPldU68 but the styles have been "lost". Modern groups are attempting to rediscover these lost arts, like HEMA https://en.wikipedia.org/wiki/Historical_European_martial_arts.
Almost everybody is familiar with Greek and Roman wrestling, but pretty much everybody everywhere has developed some kind of wrestling art. For example, the Irish developed a "Collar-and-elbow" wrestling style sometime in the 17th century, but which are traced back to various folk games played as far back sometime between 630BCE to 1600BCE and lasting until the Norman invasion. In Switzerland, the wrestling art of Schwingen dates back to at least the 15th century. Here's a video of what it looks like https://www.youtube.com/watch?v=ffRjdD78AsM You can probably imagine that this would have been practiced on some level by just about everybody involved in military matters.
Boxing has similar variation and history. The Greeks boxed, the Minoans boxed, the Romans boxed, etc. Each area had different techniques and rules. But in a military sense would have just been incorporated as part of an overall training program. Here's a fresco of people boxing (with gloves) sometimes before the 16th century BCE. http://www.historywiz.com/galleries/boxingboys.htm There's quite a bit of evidence that arts incorporating kicks were quite common. Greek Pancrase/Pankration is the best known of these due to modern revival attempts. https://en.wikipedia.org/wiki/Pankration http://www.pankration.gr/
The important thing is that arts like this have been practiced for millenia all over the West. However, even the best unarmed fighter is an easy kill for somebody with a pole-arm and a sword. So formal training emphasized weapons training over unarmed training.
As the firearm continued to grow in dominance, entire hand-to-hand combat systems basically disappeared or were ignored. A modern soldier doesn't practice his sword skills every day because just like a fight between a person with a sword and a highly trained unarmed opponent, a fight between a person with a rifle and a person with a sword would likely have similar outcomes.
Today, modern firearms training have largely replaced all other armed and unarmed combat systems in modern military training. Basically the West threw away most of the old and unworkable fighting systems given changes in the battlefield. What was left were basically "sport" arts like modern boxing and greco-roman wrestling. A few non-firearm arts (influenced by Asian arts) also showed up. http://www.theatlantic.com/health/archive/2013/11/how-to-fight-like-a-victorian-gentleman/281163/
https://www.youtube.com/watch?v=ZDNaC-2HW-A
Savate is also still popular. http://www.worldcombatgames.com/en/sports/23/savate
In Asia, due to various social, political and economic factors, modern weaponry took longer to really penetrate into daily fighting matters, and thus older fighting systems stuck around longer. They lasted so long that they ended up impacting with the modern world and being well recorded. Mediums like photography and moving pictures provided fantastic promotion. But it appears that the forms those styles took were a bit different than the modern forms. Here's a Kung Fu match from 1953 Hong Kong between two masters https://www.youtube.com/watch?v=S19VsB7__v0 Here's a more recent one at a sporting event.https://www.youtube.com/watch?v=8uKuekLXlac
edit almost forgot to mention the boxer rebellion https://en.wikipedia.org/wiki/Boxer_Rebellion
Even more important, Western military involvement in Asia reintroduced the idea of unarmed combat to military men (armed combat had shifted to bayonet training, with less emphasis on fencing and more on stabbing as hard and as quick as possible). Soldiers coming back from Japan, Vietnam and Korea brought back a revived interest in martial arts. In the 1950s for example, the U.S. Air Force introduced a Judo program. http://www.bragg.army.mil/18abc/FBCS/Documents/Basic%20Combatives%20Course%20(Level%20I)%20Handbook.pdf
It turns out this reintroduction of unarmed combat into the West has coincided with a renewed interest in "non-lethal" tactical approaches. Basically modern weapons are too deadly, turning engagements into a binary kill/no-kill fire/don't fire problem with no other alternatives. Modern Western military training has developed several unarmed fighting systems like the MCMAP which pragmatically adopts techniques from Western traditional as well as Asian traditional systems http://www.combatical.com/
Movies and Hollywood further cemented the exotic East Asian martial arts and popularized them. Nobody's made a movie of a plucky young French Knight in training using the best secrets of a German fencing manual brought to him by an old sword master to right injustice and fight the bitter and disillusioned just-returned-from-a-Vatican-prison Knight Templar in a final epic battle. But there are hundreds of similarly themed Kung Fu and Karate movies. Ninja mania in the 1980s finally cemented the role of East Asian martial arts in the public's mind and fully pushed Western traditional arts out completely.
I have a question which may be related, saving me from making a separate thread: I often hear this had something to do with the quality of steel. Is there any truth in this at all? And what was the quality of steel like? I'd really like to know.
I have an entire book dedicated to giving a factual and clear understanding of what Chinese Martial Arts really is. I have considered quoting all the relevant parts of the book but upon further consideration, I have decided that the task would be cumbersome. Due to the limitations of a reader's attention span and the amount of effort it would take to type out entire essays out word for word straight from the source, I will give only the most relevant passage in the Conclusion of the book.
"Much of the current understanding of Chinese martial arts in the West comes from the particular circumstances of late nineteenth and early early twentieth-century Chinese history, and the representation of the Chinese past in martial arts films. Most Chinese and Taiwanese people share these misconceptions about the history of Chinese martial arts, and indeed Chinese history as a whole. Very few historians, outside of a tiny group of specialists, have read the excellent scholarship in Chinese on the history of Chinese martial arts. It is therefore not surprising that when asked, the average Chinese or Taiwanese person will provide the same list of myths and hearsay that color and distort the understanding of Chinese martial arts. Familiar fabrications - Shaolin is the source of Chinese martial arts, the Daoist Zhang Sanfeng invented Taijiquan, or Chinese martial arts is about peace and self-cultivation, not violence, to name only a few - are repeated so often and so confidently that they appear to be true. They are not, and these misconceptions undermine not only our understanding of the place of martial arts in Chinese history but also the actual practice of martial arts.
A number of nationalisms and orientalisms inform the current meanings of martial arts in China and the world. The individual skill of a martial artist often becomes representative of not simply his or her abilities but also the value of the style practiced and the culture that produced that style. There is an ongoing debate over the relative qualities of the various styles, where they become stand-ins for their country of origin, adding emotional fuel to fundamentally inane comparisons. At the same time, the traditional cast of martial arts and its patina of ancient history attract students interested in connecting with an earlier period. Supposedly ancient practices rooted in discipline, good manners, and spiritual cultivation provide a cure for modern social ills. China as the most ancient of the East Asian civilizations must have the most profound of arts. The older the art, the more hidden are its true powers and value. Although China and Chinese martial arts may sometimes appear old and weak, they are in fact wise and strong. These are comforting myths for some, but myths nonetheless."
Although this doe not directly answer your question as to why unarmed martial arts seem to be more prevalent in Asia than Europe or the Middle East, I think it clears up the inherent misconception of the question. A better question would be to ask "why Asian martial arts seem more popular and prevalent in modern society?"
To which I would answer: entertainment media and the commercialization of artistic violence.
I have heard that the rise of unarmed martial arts in China had to do with a ban on peasantry owning weapons by an Emperor, and so they developed advanced unarmed fighting styles and easily concealable weapons that could be passed off as either everyday objects or farming equipment ie the bo staff, kama, tonfa, etc. Is there any truth to this?
Much as there were many regional martial arts in Asia, there were many regional martial arts in Europe. I'm not sure it's helpful to focus on unarmed arts, and I'll get to why in a moment.
Our earliest surviving manual of a martial art in Europe is MS I 33, a German work which depicts a monk teaching a student to fight with sword and buckler (a sort of small shield). This was written around 1300.
Fast forward a hundred years, and you have Fiore dei Liberi writing in Italy. He records extensive unarmed combat techniques, including unarmed defenses against a dagger, as well as sword, spear, poleaxe, and mounted combat. It's basically a how-to manual for the medieval man at arms.
He was of the opinion that all armed combat was an extension of unarmed combat, that all the principles learned in unarmed combat applied also to armed combat, and that each form of armed combat built on the others. So the sword draws on the dagger, the spear and the axe draw on sword techniques.
Fiore saw martial arts as a much more holistic pursuit than an arbitrary armed/unarmed divide. This is the main reason why I would argue that it's not helpful to compare only unarmed arts - I'm not convinced that historical practitioners would have seen this sharp dividing line.
Ultimately, a martial art is intended to kill or disable your enemy as quickly and efficiently as possible. Fiore's concept is to use whatever means necessary to do so - if you have the sword, use the sword, but don't be afraid of dropping the sword and moving to unarmed combat if that's the best way to win. Techniques often involve some of both armed and unarmed like that.
I'm less familiar with the German masters, but suffice to say that after Fiore's book, there were quite a number of combat manuals written in Germany. AFAIK, these also do not make a sharp divide between armed and unarmed arts, but I await correction from someone who knows more about the German masters than me.
The only German master I have even a passing familiarity with, Sigmund Ringeck, definitely includes a section on unarmed combat, and also includes (as Fiore does) plays which move from the armed to the unarmed. This, I feel, is again demonstrating that there isn't always a clear divide between armed and unarmed combat in the European mindset - Ringeck's martial art includes both.
In England, the main master we have is George Silver. He doesn't deal extensively with unarmed combat, but has a brief section on grapples to use in swordfighting, which again I would say is evidence that the European masters didn't necessarily see this dichotomy.
Study of European martial arts is in its infancy, but suffice it to say that there is no reason to believe that there were not numerous regional martial arts in Europe, and even within a single country in Europe.