While the short story has its roots in European myth and fable, the majority of the authors who contributed to the development of the short story as a genre of its own were Americans writing in the 19th century. The proliferation of print journals in a society that enjoyed freedom of the press contributed to this.

The short story is not an American invention! The earliest short stories appear as early as the 14th century in the works of Chaucer and Boccaccio. Authors such as Charles Perrault and the Brothers Grimm, who last time I checked were definitely not American, contributed heartily to the development of the short story.

Just because early American writers grabbed the short story and ran with it, does not make the form an American invention.

The short story did not become a distinct, fully developed literary form until relatively late in the game, in the late 18th and early 19th centuries. The works of Chaucer and Boccaccio are poems, not short stories.

The short story is an American invention simply because American authors carry most of the credit for its development as a distinct literary form.