I don't know about the legal requirements in NC or elsewhere for that matter as they pertain to leashes. I've seen signs that say riders must maintain control of their equipment at all times to prevent runaways which may imply the use of a leash, if necessary. I think by purchasing the lift ticket, you may be obligated (legally) to comply with ski patrol in matters concerning safety, but I think that probably only concerns things which present an imminent danger (i.e. unsafe behavior) and I'd be hard pressed to believe the absence of a leash constitutes an "imminent danger".
So I find more than a little hard to believe there are any state laws regarding leashes. The resort can certainly require one as a rule, though. I just have never seen any resorts which do so.