Well…
Country music, like rock and roll, has several parentages. First, and maybe most obvious, were English and Irish folk music, which were brought to the Southern U.S. by English, Scotch-Irish and Irish immigrants. American variants on the themes of these old ballads soon emerged.
Then throw into the mix the banjo, brought from Africa by slaves.
Result: By the very early part of this century, a genre known (and often derided) as “mountain music” or “hillbilly music”. It had largely the same instrumentation as today’s bluegrass, but without the rollicking rythms (more on which later).
Mountain music (Appalachian mountains, by the way) was quite popular in the South, and parts of the West, but never in the North. (The occasional square dance north of the Mason-Dixon line notwithstanding.)
Then along came Jimmie Rodgers (“the Singing Brakeman”), today widely regarded as the father of Country Music. In the 20’s and 30’s, Rodgers took country music, for the first time, into the mainstream, gaining popularity nationwide. Ironically, muuch of what Rodgers was doing was actually just the blues. Twelve bar blues, to be specific, the same form utilized by the black blues masters of the day, including the legendary Robert Johnson. Jimmie Rodgers basically took the form of the blues, and merged it with the lyrical themes of the English and Irish ballads. Rodgers also introduced some of what became standard themes in Country music: train songs, prison songs, my-woman-done-left-me songs.
Meanwhile, “folk music” was developing along a largely parallel course, also borrowing from English, Irish and African influences, though the African influence in folk music was perhaps more muted. Folk music, Country music and the Blues, along with the Gospel songs of the rural South have continued cross-pollinating each other to this day.
Among the offspring: Rhythm & Blues, Rock & Roll, Bluegrass (combining the instrumentation of hillbilly music with an up-tempo beat borrowed from jazz and swing).
And since you mention it, why would jazz be considered any more uniquely American than Country or Rock and Roll? All three borrowed, at least initially, from older traditions. You can trace jazz back along the line from modern jazz to New Orleans jazz, to blues and ragtime, through minstrel tunes and work songs and on back to Africa, in the same way you can trace the ancestry of Country and Rock.