I can definitely see where a lot of religious people would have an issue with TRADITIONAL martial arts. And I see it a lot with regard to my kwoon. The problem is, I think, that there are plenty of "sport karate" dojos, which are great for what they are, but do not represent martial arts truly. Particularly in America, Karate has become a commercial boom. These dojos have no elements of tradition, they focus on stretching/self-defense/movement. They focus on health and "reaching goals." The goals are laughable. The sensei practically guarantees a belt every 8 months. They constantly compete in tournaments (where they are wearing armor like medieval knights on horseback). The dojos are brightly lit, the gi are clean and pressed, and the environment is one of companionship and happiness. I have even seen schools that offer birthday parties for kids, homework help, etc. In such schools, there is nothing contrary to any religion, I think. (Unless you despise capitalism???)
My kwoon is traditional. When you come in to train and when you leave you bow to Sifu and you bow to the altar. On the altar there are statues of general kwan, fruits, incense, candles and photographs of the masters in our lineage. There are Buddhas and lion dogs all over the kwoon. I guess it depends on how "fundamentalist" the person is, but this bowing to the altar bothers people.
We cultivate Qi. Qi is not a "euphemism" for bloodflow or strength or breath. People who describe it thusly do not know what they are speaking about. Its a pretty good measure, because once they say that - you can stop listening, they have no clue.
We train hard. We spar without gloves, helmets, chest pieces, and all the other mess. Just groin cups and mouthpieces. Sport/American karate trains for tournaments - so they form the habit of "pulling punches" for the tournament sparring. We generally avoid tournament sparring unless it is full contact without gear. We do this because our intention is to become fighters. If we NEED to use our art, it is not for points. There will be no referee. There is no "honorable" fight - we fight to destroy our opponent effectively and efficiently. Many religious people would take issue with this mindset - it does not teach "turn the other cheek" or "fighting is immoral." There is no pacifism here.
Finally, select elder students are selected, after years of service, to become Sifu's disciples. There is a ceremony. They pledge their life to him and his teachings. Again, for those used to the shallow American Karate schools, this is difficult. And I can certainly see some religious people being very uncomfortable with this relationship with a sifu.
We break bricks. We do not break pre-manufactured boards. We do not use spacers. It is not a matter of "power" alone. But there is no "evil spirit" that zaps from our palms.
I guess it REALLY depends on what people mean when they say "martial arts." Do they mean the sport stuff? Or do they mean the traditional stuff?