I kid you not. There are literary writers and there aren't. Literary is different from genre just as art is different from craft. As in painting and sculpture and music, there are craftsman and there are artists. King is a master craftsman, not an artist.
Literary works (novels or short stories) are not characterized by language, as is commonly claimed, but by unique insights and images created by the language. As in painting and sculpture and music, these are used to reveal universal connections. To discern and describe connections not before identified is what an artist, and a scientist, and a philospher, does.
Literary writers seldom write genre because they are almost always consumed with the investigation of the here and now. On occasion when they do write using vehicles of mystery or historical, supernatural or science fiction, they are still writing literary works that may appear to be genre, but aren't. They're literary in genre skin.
It isn't necessary to go back to the "Classics" to find works of literary fiction, because many of them weren't. "Literature" is made up of both literary and genre. There are many quality literary writers active today. Their product, though, may require some digging to find.
The above is my personal opinion, I may make no claim to know any universal truths about ths, or anything else.