The least populated state in the US is Wyoming with less than just 493,782 people. The the city of Washington, D.C. has more people than that all by itself. I happen to live in the state of Illinois. When I mention that to people not from Illinois, they tend to immediately think of Chicago. It is a fairly good-sized city, once the 2nd largest in the USA (now third) with a population of 2,896,016 people. That is roughly the same as Wyoming, Vermont, Alaska, North Dakota, and South Dakota all added together. In addition to those 5, Delaware and Montana also have less 1 million people living in them.
But even when you in a fairly large state Illinois with a good sized population and a world class city like Chicago, there is still a lot of empty space most places. Illinois is much more than just Chicago. Most of Illinois is also farmland, just like North Dakota. Our farms are not quite as large and we have several cities (not just Chicago) larger than North Dakota's largest; nevertheless, most of it is NOT urban. Every time I drive somewhere, I have to be careful about deer being on the road. It is nothing to also see raccoons, possums, fox, or coyote. When I used to live along the Mississippi River, I could watch eagles fly overhead from the couch in my living room.
There was a TV show on years ago, MASH, in which the actors used their real hometowns as the hometowns for their characters. One of them, McClean Stevenson, was from Bloomington, Illinois. I remember when his character talked about how small it was because to me it was "the city" -- about 60,000. It took us an hour to drive there, but when we did there were restaurants and movie theathers, and all sorts of things that we never even dreamed of. Eventually I even attended university in that town. But I have to admit, it is still just an overgrown version of other small Illinois towns. It doesn't think like most cities think.
Today, I live in a county seat, the principal governement municipality in our area. And the town is still so small that we have exactly 2 groceries, 3 gas stations, and 5 stoplights.
You asked if there are other rural areas in the US. Well, I know that Woodrow said that North Dakota is the least densely populated state in the USA. I think that Wyoming is the least densely populated state. It might depend on which census you look up. But I can tell you that last summer we drove across it and I went 20 miles (32 km) without even seeing another car, let alone a town. Southern California and the east coast from Boston to New York to Philadelphia to Washington is one gigantic city, even including open spaces it still averages out to more than 1100 people per square mile (450 per square km) -- the cities themselves are 10X that. But once you leave those two coasts and start moving inland you are looking at an average that is less than 1/15th of that even with cities like Chicago thrown in.
In fact, just to give a sense of how the USA really is two different worlds, our largest city, New York City, is only 13th on the list of world cities; but even still, just as a city, it is larger than the population of all but 8 of the 50 United States.