If they didn't keep your transaction open, they wouldn't be able to charge you for damage or incidentals. It's why hotels require credit cards for bookings in the first place. Usually they haven't seen your room when you check out.
Unless they saved the credit card number, either directly or through vaulting at their provider. Both of those are easy and common, and the second one is even safe (since it only allows that particular merchant to charge the card at will, it doesn't appeal to thieves like an actual credit card number would).