MMA is based on the base skills of boxing, wrestling, Muay Thai, and Brazilian Jiu-Jitsu (BJJ).
Boxing has a long tradition as a Catholic endeavor. (I learned to box in the old Catholic Youth Organization amateur boxing program, and my coach was our parish priest.) The succeeding waves of Catholic immigrants to America (Irish, Italian, Mexican, Filipino, etc.) traditionally had a lively interest in the sport. Catholic colleges have traditionally fielded strong boxing and wrestling teams. As a sport, if done with due concern for your opponent's safety and with good sportsmanship, there is no problem with it for a Catholic. If anything, MMA is safer than boxing as one can tap out if at risk of injury, whereas a boxer usually cannot quit in the middle of a match. The MMAist's ability to change level and shoot in for a takedown also makes the sport safer, as one can change the fight to grappling if one is soaking up too many punches or kicks.
BJJ is also a great and fun sport (I've been doing it since 1996, off and on), which originated in the largely Catholic country of Brazil (most Brazilian teachers I know are Catholic). It lacks the mysticism of many Asian martial arts, and is based on scientific principles of balance, leverage, and technique. Its philosophy is based on the traditional western sports principles of hard work, conditioning, respect for your opponent, and sportsmanship, not badly recycled Shintoism or Buddhism. As a sport that favors grappling, it allows for someone who is attacked to defend himself with a minimum of damage to the opponent by taking him to the ground and controlling him with a legally (and morally) defensible use of force by pinning, controlling, or applying a submission hold. I think BJJ is much better for children to learn than, say, Tae Kwon Do, as the zero-tolerance policies in schoolyards will punish a child severely for striking back against a bully by punching or kicking him, whereas with BJJ a child can take a larger bully down, obtain a superior position on the ground where he can control the bully and avoid injury, and wait for a teacher to separate them.
It's likely than St. Paul was trained as an MMAist, by the way. Although Jewish, he was sent to a Roman gymnasia by his dad, where the usual course of scholastic instruction included wrestling, pugilism, and the pankration, the Greek "all powers" style of MMA. Paul scatters sports metaphors throughout his epistles (especially foot-racing), and seemed to take some pride in his athleticism. The Corinthians were fond of the Isthmian games (which included Pankration), and like any good preacher, Paul used home team sports metaphors when addressing his flock. He seemed to refer to the shadow-boxing of boxers when he said, “I do not fight like a man beating the air. No, I beat my body and make it my slave” (1 Corinthians 9:26)."