So, if it's a privately owned facility, they can suspend a coach for no reason. Just because... Correct?
No facility would suspend a coach without good reason. It's poor customer service and damages their goodwill. Plus, it requires effort and most facilities don't care as long as it doesn't interrupt their income. If you think your rink was that flippant, you're as much in denial as your coach who's claimed that "It was just a simple discussion." Something happened (raised his voice, cursed the coworker out?) and the suspension is in reaction to that something.
I worked at a privately-owned rink where a father loudly threatened (in the crowded lobby) to kill the figure skating director because the man's daughter took third place in an in-house LTS competition. When confronted, the man insisted he never said that, but the hockey director was in the crowd at the time and had seen/heard everything! Management had the man escorted from the rink and the family was told that the kid could continue to skate, but the father couldn't come on the property or they'd call the police.
I'm sure there's more to the story than what you are aware of or have told us; there are
always three sides to every story. From what you're saying, you didn't witness this; you're only hearing your coach's side of the story. My best advice is to MYOB and address your skater's coaching needs. There ARE other coaches at the rink, ask your former coach where he's coaching and decide if you want to go there for lessons, and see just how professional your coach is in dealing with this conflict. If he's not addressing it, he probably did contribute to the blow-up and is in denial.
Try to stay neutral and open-minded so that you don't ruin your own reputation or offend others in your earnest desire to defend your coach. Do not discuss, gossip or start a petition. It is your COACH'S responsibility to clear his name and resolve the situation professionally before his career ends up in the toilet. I've seen spats over coach-hopping students result in a coach being banned from a rink. They go elsewhere, but they can't "take" their students with them; although some families can choose (unsolicited) to change rinks if they're that committed to the coach.
That said, any private business can refuse to serve a customer: dry cleaners, restaurants, markets, etc. Skating rinks are no different in their rights.
As a SafeSport Chairperson, I can honestly say that there is a new atmosphere of "zero tolerance" for coach's misbehavior. Coaches need to realize that the times are a'changing and a new era of absolute above-board professionalism is now required.
I think Skittl makes a good point: go to the office and, without asking for details, tell them that you want to know when your daughter's coach will be returning. If the answer's "never," indicate that you will have to take your business elsewhere since you're committed to that particular coach above all others. There is something to be said for loyalty, but I will say that IME, most people who make threats like that never follow through. They just meekly swallow their pride and end up with a different coach at the same rink.