alt Hacker NewsPoetry has fallen out of favor in culture at large, so I think it’s more logical to expand the question to authors more generally.
In that case, I think The Great Gatsby is the most logical equivalent to the Iliad and Odyssey. To me it conveys the essential elements of the American character: obsessed with reinvention via commerce, but inevitably tied to the past. It’s also probably been read by the majority of Americans, or at least the story is well-known. Can’t say the same for Twain, Melville, etc.
For more recent authors though, I think Don DeLillo is the best chronicler of mid-late century America.
> It’s also probably been read by the majority of Americans, or at least the story is well-known.
Read by, yes. Story well-known, I doubt. What is the story of The Great Gatsby, in a few words? (But then again, what is the story of the Iliad, anyway? Something about the wrath of Achilles, right?) [I should clarify: to answer that the Iliad is about "the Trojan war" or "Achilles dragging the body of Hector" seems insufficient to me in the same way it would be insufficient to say that Gatsby is about "a guy who throws parties" or "a murder in a swimming pool."]
Twain is the obvious "American Homer" not just because his name pops immediately to mind, but also because his two best-known works (Tom Sawyer and Huckleberry Finn) stand in the same relation to each other as the Iliad and Odyssey: a minor character in one becomes the protagonist of the other. As someone above already commented, too, both works are episodic and have given us many pop-culture images: I dare say only Tom's painting of the fence is proverbial, but the cave adventure, the feud, attending one's own funeral, and probably several more episodes are embedded in American pop culture to the same extent as the Cyclops, Circe, and the Lotus-Eaters. Gatsby is a great novel, but what has it given to our collective unconscious? The green light and... beautiful shirts?
What does it say that there have been three adequate film adaptations of Gatsby, while there have been (as far as I know) none at all of Tom Sawyer, Huckleberry Finn, the Iliad, or the Odyssey?
(Side note: About 15 years ago I read Fitzgerald's The Beautiful and Damned and thought it was very good and inventive; I recommend it. These days Wikipedia tells me that "many Fitzgerald scholars typically consider The Beautiful and Damned to be among the author's weaker novels," so maybe something excellent is waiting for me if I ever get around to reading This Side of Paradise or Tender is the Night.)