I'm assuming it has something to do with the geography, and perhaps the first Europeans to venture to America assuming they were in India? If this is so, why did the term stick for so long, to the extent that everyone still thinks of "cowboys and Indians"?
Colombus was searching for a fabled western route which would end in India, and so initially he believed he had stumbled on "West India" which is why we call that area today the West Indies.
He wrote several letters back which became circulated, in which he used the term "Indian" in reference to the people he discovered there. The one announcing his discovery was the first letter in the Old World referencing what he had discovered, and so was precedential.
I heard from other Indians I had already taken that this land was an island, and thus followed the eastern coast for one hundred and seven leagues, until I came to the end of it.
http://www.ushistory.org/documents/columbus.htm
The term stuck simply because of inertia. Once everyone heard of a fabled new land peopled with "Indians," coupled with the limited understanding of who those people were and their own history (they were perceived as little more than savages), meant European people cared little for distinguishing them.