The short answer: yes. They didn't fade into obscurity either; they became major international sports that continue today: boxing and wrestling.

With respect to the development of unarmed martial arts, one very important point is that most martial arts involve the use of weapons, and unarmed martial arts are typically only important in non-military contexts. Historically, the most important non-military setting for unarmed martial arts was sport.

Both unarmed and armed sporting martial arts have developed. The major types can be described as boxing (focussing on unarmed striking, although techniques such as clinching, throwing, tripping, etc. might be allowed), wrestling (focussing on grappling or throwing, although some striking might be allowed), stick-fighting, and marksmanship (archery, target-shooting with guns, etc.). The first two are unarmed, and the last two are armed.

Wrestling has very old roots, with clear evidence of its practice in ancient Egypt and Mesopotamia, and the appearance of wrestling in myth (e.g., prominent wrestlers such as Heracles, Bhima, Gilgamesh) suggesting that it's much older than surviving written records of real-world wrestling. Europe enters the wrestling scene very early, with wrestling becoming an Olympic sport in the 8th century BC. In the Greek world, wrestling was taught by professional instructors, and successful wrestlers could obtain both fame and fortune. Although it lost its status as an Olympic sport with the disappearance of the ancient Olympics, wrestling continued as a popular sport in many parts of Europe, with many regional wrestling styles surviving to the modern day, with some forming the basis of modern Catch (Catch-as-catch-can) and Greco-Roman (which is a 19th century style, without any direct connection to ancient Greek or Roman wrestling). Wrestling was also a major martial art/sport outside Europe, with traditional styles surviving into modern times in China, Mongolia, Korea (Ssireum), Japan (Sumo), India, Iran, Turkic Central Asia and Turkey, and Africa.

Boxing also appeared very early in Europe. Perhaps the first clear evidence is the Akrotiri Boxer Fresco from Minoan Crete, early in the 2nd millenium BC. Like wrestling, boxing became part of the ancient Olympics. While boxing was often a lower-class cousin of wrestling, traditional styles survived in Europe into modern times, and were the foundation of modern boxing.

Combing both striking and wrestling: Pankration (literally, "all powers"), ancient Greek MMA (mixed-martial arts), allowing almost all techniques (only biting and eye-gouging were prohibited, and not even those in the Spartan version). Unlike boxing and wrestling, Pankration did fade into obscurity. (However, some Early Modern boxing styles allowed significant amounts of grappling and throwing, headbutting, chokes, etc.)

At the same time that boxing and wrestling were developing into their modern forms (in about the 18th century), we see unarmed martial arts assuming a more prominent role in China (compared to the earlier dominance of armed martial arts). This process began in China in the 16th century, and parallels the rise of the handgun (arquebus, musket) as the key military weapon. This began earlier in the Ryukyus (i.e., Okinawa), with the unification of the Ryukyu Islands in the 15th century and restrictions on weapons and weapon ownership by the new central government, and much later in Japan and Korea. East Asian unarmed martial arts, apart from traditional wrestling styles, are largely an Early Modern and Modern phenomenon, although with older roots. Why, we might ask, didn't this happen in Europe - the development of unarmed martial arts styles other than wrestling and boxing? Noting that 18th century and earlier boxing often allowed a wide range of techniques (head-butting, chokes, throws), about the only "martial arts" thing that's missing is kicking. But kicking didn't disappear from European martial arts with Pankration - it continued to be taught and practiced, and appears in the unarmed portions of Medieval European fightbooks. In the 18th century, we find a range of kicking techniques being used in fighting in Spain, Portugal, and France; these formed the basis of Savate, as developed by Michel Casseux, who opened his Savate salle (i.e., dojo) in 1803. He systematised the techniques and their teaching, and made them more suitable for sport than the street-fighting they previously appeared in.

We might ask why boxing and wrestling are often ignored when considering what is a martial art. If they are recognised as martial arts, the question of whether there are European martial arts would not arise: boxing and wrestling would be the obvious answer even before the question is asked (the Anglophone world ignoring such French things like Savate). European unarmed martial arts are today almost purely sport-oriented, which might be a partial explanation. However, much of the survival of Asian unarmed martial arts into the 20th century and beyond is due to their growth as sports and physical education (e.g., Judo becoming an international sport, and Karate becoming part of physical education in Japanese schools).

Further reading:

On Greek wrestling, boxing, and Pankration: Michael Poliakoff, Combat Sports in the Ancient World, Yale University Press, 1987.

On the history of Chinese martial arts, from ancient to modern times: Peter A. Lorge, Chinese Martial Arts: From Antiquity to the Twenty-First Century, Cambridge University Press, 2012.

On Savate - I don't know of a good book-length work, but a brief overview of its history can be found in the "Savate" entry in: Thomas A. Green, Martial Arts of the World, ABC-CLIO, 2001.

On the survival of Chinese martial arts as physical education: Brian Kennedy and Elizabeth Guo, Jingwu: The School That Transformed Kung Fu, Blue Snake Books, 2010.