IINAL, but I think your friend might actually have a case, if there are managers who rarely or never work on weekends. If he can show that in court, he could make a case that the “must be able to work on Sundays” catch was created solely to prevent him from qualifying because of his religious beliefs.
On the other hand, if he does file suit, he will probably earn a reputation of being a “troublemaker”, deserved or not, which might make it impossible to continue working for the company, and might make it difficult to get hired elsewhere.
So filing suit, or even mentioning the possibility of doing so to management, would almost guarantee he won’t become a manager there.
But there might be more at work here. The district manager might have decided against promoting your coworker for some reason other than religion. Maybe he thinks that your friend won’t work well with the other managers, but just made up the excuse he gave to avoid looking like he was setting up a good-old-boys clique. Or maybe your friend hasn’t brown-nosed enough. Or it could be something else entirely.
Does your coworker have a good relationship with one or more of the current managers? If so, maybe he could talk to them about what’s going on. They might have more information about the situation. If enough of the managers think the district manager is being a jerk, there’s the possibility of going over the d.m.'s head. But without some support from lower management, going over the d.m.'s head would probably bring nothing good.
So, in my ever-so-humble opinion, your coworker should talk to a Real Lawyer about the situation, and should figure out what he really wants. Does he just want the district manager to change his mind? Does he want to prove a point?
Of course, there’s always the ultimate revenge: quit, get a management job at the competitors, and go a good enough job to pull customers away from his current employer.