This is a perennial question to which there can never be a definitive scientific answer. To understand why we can never know this, we need to think about how you would go about proving this one way or the other. It would be necessary to take two people who are genetically absolutely identical and raise them in different ways. There are attempts to do this through the study of twins, both identical and fraternal. However, we do not completely understand how much genetic difference there is between twins and so we cannot truly understand how much of a role “nature” and “nurture” play respectively.
That said, it is quite clear that both nature and nurture play large roles in our development as human beings. They each have an impact on our personalities. They each have an impact on how intelligent we are. This much is clear. The problem is that there is no way to reliably and completely untangle the effects of our genes from the effects of our upbringings. Until (or unless) science progresses far beyond what we can imagine today, this is a question that humans will never answer.
This question is one of the classic unresolved questions in science and philosophy. In Western thought, it has had several different iterations, with Christian thinkers positing the inherent depravity of man, and later empiricists like John Locke arguing that humans were born with a "tablua rasa," or a blank slate. The modern debate involves less reductionist positions, and generally tends toward the conclusion that both factors play pivotal roles in shaping human development. Anthropologists and other social scientists stress the importance of cultural factors in shaping human development while conceding the importance of genetics. On the other hand, scientists have in the last two decades begun to seriously consider the extent to which cultural development is itself limited by genetic factors. In other words, culture is created and reproduced in social contexts that are shaped by the fact that people are genetically "hard-wired," if you will, to be social creatures. Ultimately, however, most would concede that while humans face significant genetic constraints, (or "nature,") the way that people have responded to them is fundamentally cultural (or "nurture.")