Not Smetana directly, but his fellow Czech successor Dvorak’s US visit in 1892-1895 was a catalyst for American composers turning to American home-grown musical idioms for their inspiration (as opposed to exclusively aping European musical styles). This led to the flowering of “American styles” in classical and other genres shortly after that. In fact, Dvorak’s first batch of students taught Gershwin, Ellington, among others....
Dvorak went deep int the American heartland, spending months in Spillville, Iowa, where there is a small Dvorak museum along with a huge collection of carved clocks.