I feel compelled to point out that your question is based on something of a false premise. While East Asia certainly has a rich martial tradition, the West and Europe does as well - we just don't tend to immediately think of them as martial arts because the Eastern classics like karate, kung fu, etc. play such a large role in pop culture. Despite that, fencing, jousting, Greco-Roman wrestling, boxing, savate, and cane fighting are all examples of Western martial arts that are comparable to Eastern martial arts in terms of formalized technique, training methods, and competition.