It needs to be more widely understood that the horse's status as a native North American species is beyond serious question.
A "native" species, in evolutionary terms, is defined as one that differentiated or diverged from its immediate ancestor species within a specific geographical locale. The contemporary wild horse in the United States is recently derived from lines domesticated in Europe and Asia. But those lines themselves go much further back in time, and converge on populations that lived in North America during the latter part of the Pleistocene (2.5M to 10k years ago).
The morphological (fossil) evidence and the more recent DNA evidence (although preliminary), points to the same conclusion: the species Equus caballus—the species encompassing all domestic horses and their wild progenitors—arose on this continent.
The evidence thus favors the view that this species is "native" to North America, given any rational understanding of the term "native". By contrast, there are no paleontological or genetic grounds for concluding that it is native to any other continent.
From a scientific standpoint, it is completely irrelevant that native horses died out in North America 10,000 years ago, or that later populations were domesticated in central Asia 6000 years ago. Such considerations have no bearing on their status as having originated on this continent.
Reintroduction of horses to North America 500 years ago is, biologically, a non-event: horses were merely returned to part of their former native range, where they have since prospered because ecologically they never left.In response to a Wall Street Journal Article in 2012 by Stephanie Simon contending that Wild Horses are not Native to North America, Ross Mac Phee wrote the following response:
Ross Mac Phee PhD – Wild Horses Native to North America