Isn't the definition of urban = a city. No, Providence is not Boston or New York. It is a SMALL city.
I lived there for five years and it really grew on me. Mostly, I like the fact that you can get pretty much anywhere in under 15 minutes, even at rush hour. (As opposed to Boston, where I live now and you can't get ANYWHERE in 15 minutes.) Obviously, because it's small, you don't have all of the cultural opportunities/nightlife of a big city, but there is the RISD art museum, excellent bands at Lupo's, and a great variety of cuisine (although without that many Chinese or Mexican options) PLUS a reasonable cost of living. And, if you really want the big city nightlife, you can go out in Boston, as my roommates liked to do.
I would say the only downside of the experience you would get at a hospital there is that you might see less of the really unusual cases than you would expect for an area with that population, as many people do go to the famous hospitals in Boston for treatment.