alt Hacker NewsDefinitely Twain and it's not even close. Huckleberry Finn even has the episodic flavor of The Odyssey.
For books you have "The Innocents Abroad", "Roughing It", "The Adventures of Tom Sawyer", "A Tramp Abroad", "The Prince and the Pauper", "Life on the Mississippi", "Adventures of Huckleberry Finn", and "A Connecticut Yankee in King Arthur’s Court".
That's a hell of a lineup and includes a LOT of the flavor of the US as a young country.
I don't see how it's not him, Whitman (the timing of his work is a strong point in his favor, being so concerned with the civil war and that apparently-neveaending struggle in America's metaphorical soul), or maybe Melville though I think he's the weakest of those three. Most other plausible answers are too late in the country's history to take that particular title, for one thing. Irving and Poe are early enough but I don't think either's even close to the combo of clout, staying-power, influence, quality, and breadth of coverage of, specifically, Americanness that one would expect for this (and I like both of them!).
I would agree with other posters that if we allow the search to stretch into the 20th century, odds are we'd end up with a filmmaker or songwriter, but I think we've got our all-around best options already, and they're all (basically) 19th century.