Europe actually has a fairly rich history of armed and unarmed martial arts, and there is in fact an entire community of Historical European Martial Arts (HEMA, sometimes also Western Martial Arts) enthusiasts, some of them who are on reddit in subreddits like r/wma and r/hemascholar.

So for example, Western martial arts would include various styles of boxing, wrestling, kick-based martial arts like savate, various schools of fencing (which covered a number of different weapons, not just saber, foil, and epee but also bayonets, daggers, zweihanders, sword-and-buckler, sword-and-dagger, sword-and-cloak, polearms, etc.) - Paulus Hector Mair even includes dueling with scythes, but that's probably ahistorical; stick-fighting (as in the Irish Shillelagh Bata, and the Great Stick, etc.), archery, and so on.

So it's not that Europe didn't develop a culture of martial arts - and there are many parallel developments in European and Asian martial arts (for example, there developed schools and lineages of instruction, and even manuscript traditions). However, the two did develop largely independently, and there are some distinct differences. Historical European Martial Arts, for example, has no direct equivalent to qigong in Chinese martial arts, or equivalent principles in other Asian martial arts.

The biggest difference, however, is that by and large many historical European martial arts effectively died out by the late 19th century. The great fencing schools of medieval and Renaissance Europe left a rich tradition of fencing manuals that give us an idea of the breadth of teaching and techniques available, but the actual institutions largely became defunct as swords, polearms, and other "martial weapons" ceased to have a primary use on the battlefield as firearm technology became more abundant, powerful, and accurate; and dueling and trial by combat became restricted and finally outlawed in civilian life.

Boxing, wrestling, archery, and fencing became increasingly "sportified", although this was very much a late development - for example in the late 18th and early 19th century you saw the development of distinct (and competing) national boxing styles in France, Great Britain, and the United States, which you can read about in “A Straight Left against a Slogging Ruffian”: National Boxing Styles in the Years Preceding the First World War.

It should also be understood that this "die out" of martial arts practices was not unique to Europe; in Japanese martial arts for example, Sōjutsu (spear-fighting) largely died out, though some of its techniques were later adapted to the bayonet, and it is still taught in some schools.