They already do. Muslim women are not allowed to wear a head-dress that covers their face when taking pictures for photo ids, and in many public job roles including teaching.
Read the supreme court decisions. Businesses cannot discriminate on the basis of gender, race, religion, disabilities, and other protected grounds. At this point sexual orientation is not a protected ground, but it is foolish of a business to discriminate on this basis. As noted in the court decisions you cannot use your religion as a claim basis to not serve business customers... otherwise every racist would be using their 'religion' as a claim to not serve minorities in their business.
This issue of muslims refusing to serve people because of claimed religious grounds has actually happened. After Bill clinton's disastrous Somalia episode, eg Black Hawk Down, his administration foolishly granted refugee status to large number of somali muslims, who were settled in Minneapolis of all places. They have done little but cause trouble in the intervening years. Many work as cab drivers. They have asserted a right to refuse to transport passengers carrying alcohol or who are intoxicated, even though as licensed common carriers they are legally obliged to do so. They also refuse to transport dogs, meaning they refus e service to blind people with guide dogs. It finally came to a head at the local airport, where a large percentage of the drivers were sosmali muslims. Blind people with ddogs were having trouble getting rides. Cabbies were demanding to know if riders had liquor in their bags. The airport authority finally put a stop to it with threats to suspend any drivers who refused service, but the muslims took it to court, of course. http://www.mprnews.org/story/2008/09/09/muslim_cabs_court See eg http://www.nbcnews.com/id/16472393/ns/travel-news/t/booze-dogs-too-much-some-muslim-cabbies/ . I draw a distinction between someone with a government license for common carriage and someone running a totally private business. Anyone who wants to can open a photography shop or a bakery. The government however typically limits the number of licenses for cabs. In return, the licensees have an obligation to charge regulated rates and serve all comers.