Used to live there. New Hanover is the only hospital in the city itself. It's a pretty big hospital, I think somewhere between 800 and 1000 beds, accredited with an ER. Pretty strong reputation, here's the site: https://www.nhrmc.org/.
I think the only reason you'd need to go somewhere else is if there was a very specific medical concern, in which case some of the hospitals in the triangle area may come into play. ~2 hour drive