If you're okay with taking Yes-Man's path, you can basically wait until the very end to kill Mr. House. There is no need to take him out any sooner than the very absolute end.
The Legion and NCR, respectively, will want him out of the picture much sooner.
So, there's two seperate questions that could be answered here.
Can you fight unarmed, using primarily punches, and wear boxing gloves? Absolutely. In fact, Unarmed is among the stronger combat specializations in the game, though Boxing Gloves are somewhat weak specifically, because they cause only subdual damage - meaning they will knock your opponent over, but not kill them. Useful sometimes, but rarely.
Can you fight in a 'boxing match'? Well, not per se, however, there are two opportunities for you to engage in a sporting fight for some caps. The Thorn in westside is home to one such opportunity, where you'll have the opportunity to fight off various monsters of the Wasteland such as Radscorpions, Giant Mantises, Deathclaws and Cazadors. Any weapon is fair game in the Thorn, though you're welcome to go in unarmed if you're looking for a challenge.
Male (and only male, the legion are nothing if not chauvinistic) Players aligned with Caesar's Legion meanwhile, have the option to fight in The Arena, located at The Fort. In the Arena, the only weapons permitted are Machete's. All your other weapons are confiscated at the beginning of a match, and you are of course free to use your bare hands instead if you so desire.
Best Answer
For Yes Man, you will have to finish evaluating all the different factions. He will give you further tasks to complete once the factions are done.