Jazz radio WNCU Durham 90.7 has a tag-line for their radio identification. It's "90.7, WNCU. Jazz...America's indigenous music."
Indigenous music, to my mind, belies a traditional music, of a reasonably homogeneous ethnicity or culture. See Wikipedia's take.
Jazz as it's currently known certainly has formative roots in the U.S. South. But jazz has gone through many many developments and has been fervently and skillfully adopted by many ethnicities, cultures, and locales.
I'm not trying to split definitional hairs here.
My point is that if there is something about jazz that is uniquely "American", it's that jazz, like the U.S., is eternally re-making its own image. Jazz, like the U.S., is not homogeneous. To me, the U.S. is a nation that was built initially on invention, and has never stopped re-inventing itself. As such, it is eternally Modern in the artistic or literary usage of the word. Jazz deconstructs sound, if only to rebuild itself. One need only listen to the music of Thelonius Monk, John Coltrane, or Sun Ra to get a flavor of this. Some say jazz in 2010 is static. Innovation has stalled, its rhythms have been incorporated into rock and roll, hip hop, electronic music, and so forth. There is no need to re-invent bop, or cool, swing. It's all been done. Who was the last jazz musician that was a household name? To this end, jazz, as it becomes more static and reiified in its form and content, is becoming more indigenous all the time. So maybe WNCU is onto something. Let's just hope that the U.S., similarly, has also not lost its ability to be modern.