Neither actually. In fact, the bow predates the existance of Europeans and Native Americans. As of right now, the earliest evidence we have for the use of a Bow and Arrow appears to be from South Africa roughly 64 000 years ago. This is well before the first definitive evidence of human settlement in Europe(.c 45 000) and far ahead of the earliest estimates for humans in the Americas. The Bow is likely one of the technologies that Homo Sapiens brought with them out of Africa, or at the least one that spread quickly after its invention as evidenced by the fact that nearly all human cultures have utilized it. (Slight tangent here: I say nearly, because Aboriginal Australians did not have the bow. The reason why is really up for debate. It's certainly possible they were isolated before its invention/ spread. But there are many other reasons why it might not have been used there: Ill link a very interesting askhistorians thread on the subject from a couple years ago here if you're interested.