Game of Thrones/A Song of Ice and Fire is arguably the medieval/ancient world setting most familiar to Americans, and what's more, is one of the few iconic fantasy series written by an American (and one who's always had an affection for comic books). To a certain extent, the series is very much Medieval Europe through an American's perspective in the same way Biblical stories and and 1001 Nights are the Eastern world through the view of a European.
I think the franchise fills a lot of the check boxes for a piece of popular culture that would endure in a Medieval American culture--it's homegrown, its setting is pre-Industrial (so no "scientist=wizard" metaphors that I would like overdo) and it is just about as iconic as anything that has become popular in the last ten years. Its major hurdle, one supposes, is that the large array of characters and plot points might be a little dense for a world with limited resources and literacy, but I would definitely keep that near the top.
It might be a little fun to think about the various aspects and compare and contrast them to Medieval America.
Martin has gone back and forth on how long the continent of Westeros is, but the general agreement is that the Seven Kingdoms runs about 3,000 miles longway, which isn't too different from the length of the United States (about 2,6000). If you cut off the North (Which, spoilers), and then the area west of the Mississippi, you have two areas of comparative size, and according to some estimates, a comparable population. (The Seven Kingdoms is estimated to have something between 25 and 50 million, a smaller density than the Eastern United States, which makes sense, as the American Heartland has less mountains, desert, etc) If you get in the the Western U.S. you get a more diverse climate, and biomes that are also found on the continent of Essos. This will be important in a bit.
Urban-wise, the continent of Westeros has five major cities. America has " a big seven", although both the below-the-neck area and the American heartland respectively, have four, which is an interesting coincidence. However, no Game of Thrones lore makes any population figures for any urban settlement besides King's Landing, which is about 500,000. (Which is probably larger than city in Medieval America.) The map has many spots for "towns", while White looks to be staying away from showing settlements smaller than 25,000. (The United States, despite being a mercantile republic, does not seem to have any of the largest cities) So it's hard to compare the two on that front, but America is clearly more urbanized. It should be said that the 7 Kingdoms are almost completely feudal, while America has a wider variety of governments, especially out west.
In fact, one can compare the western half of America to the continent of Essos, which is much more rife with city states. If one doesn't want to cheat too much, one might include the Free Cities as part of whole "area" as comparable with the U.S. (Basically quantifying "The North" and beyond as the equivalent to Canada and Alaska) you have a lucky 13 cities, which as it turns out is the exact number of the larger urban centers on this map here. Should one want to split the difference between the Seven Kingdoms and the Valyrian Freehold, you would get five cities, which is, interestingly, the number of THE most important cities according to the trade maps.
One thing that George R.R. Martin, and many cartographically -inclined fans focus on, is the fancy castles. White does have a few "major fortresses" on his map, but much, much fewer than any Game of Thrones maps. It should be said this is possibly a difference in world-building, as Martin gets a little more into the fantastical, creating many expansive fortifications that stretch credibility, or are found on geological structures that simply don't exist in America. (It's possible the U.S. terrain isn't especially suited to castle building, but we'll never know) In Martin's world, castles are seats of power unto themselves, and capitals are less often than not part of the larger mercantile areas. On the subject of capitals, the one thing White has that's unique to the Medieval American world are the District Supervisors, which are located in former capital cities, and get marked on the map regardless of size. How Supervisor cities differ from other settlements in terms of infrastructure is up for conjecture.
On another note, somebody came up with a few trade maps of their own, which varies a little bit in terms of resources, but there are similarities as well.