That's why Mongolians are also so successful in pro sumo, unlike other countries. They have very long and deep tradition of wrestling competition.
It is very difficult to determine which is the oldest "living" (still in use) martial art, but boxing is certainly not it. It is difficult because of the vast majority of falsified history, and many other reasons. Wrestling was an Olympic discipline in ancient Greece, that's for sure. Kung fu is a very old martial art and is now in use. Boxing is certainly not the oldest martial art, and to determine which one is; for that, we need a historian more than a connoisseur of martial arts. And the second thing is that this martial art is still "alive" (in use), and kung fu is.
There are african tribes that seem to have intact martial arts (often involving some sort of stick or club) who live much as they did thousands of years ago. I always assumed something as primal and intuitive as sporting combat would have grown out of training for war and then applying that training during peace for entertainment (and probably gambling) far before the invention of things like agriculture or the written word.
Boxing definitely goes back to ancient Greece, though it would look different from what we say today. Wrestling/Grappling is much more natural for humans overall however the punch tends to be most effective in combat as a punch to the head can literally end a fight in a second or less turning off the human brain aka computer.
Wrestling has to be the oldest form of human fighting. Look at the "combat" of apes and it sort of approximates rough grappling. Look at todlers even when they try to get physical. If going by specifically written tradition, boxing, wrestling, pankration and other forms all go back to Greek antiquity. Pankration lives on more or less in modern MMA.
There is the issue of whether mock combat, with some sort of limits to violence, constitutes a martial art. If the answer is yes, then it is not even limited to humans. Wrestling is probably the earliest form of combat sport, and it dates back to at least 3000BC, in what is today Iraq. Even before that, there was probably some kind of organized military training, that would have amounted to a martial art.
I'm not a specialist but isn't the Greco-Roman wrestling that exists today not the direct lineage from Greek wrestling? Using that standard modern Boxings lineage can't go back much further than the Queensbury rules as ungloved boxing before that period was very much its own entity compared to today.
The ancient Greeks had boxing, wrestling, kick-boxing, and pancreation, or "all forces," which was what we would call mixed martial-arts. Boxing and wrestling are indeed mentioned by Homer in the Illiad as being events in the games Achilles holds in honor of Patrolocus.
Again, most ancient sports don't have unbroken lineage. How would you compare boxing to sumo? The first banzuke that exist are from the 18th century and the first legendary yokozuna lived in the 17th century (though his historical existance is disputed). I am not sure we can talk about the boxing from the 17th century, but I am not an expert.