For the Greek's the answer is that Oedipus was hemmed in by his fate. His inquiry into the truth of the prophecy (and his origins) serves only to reveal his guilt, his moral situation; his crime.
If he had turned away from the inquiry, he would still be living in the same situation, married to his mother. It was only a matter of time until his punishment would come, along with the revelation of the truth.
Dramatically speaking, we must have both the fate and the inquiry however. Without both of these elements, the play would become action-driven by necessity and would then lose its qualities of intellectual investigation into the nature of guilt, fate, and the inescapability of conscience.
This is an important question. In our modern world we like to have it only one way. Was it fate or Oedipus' sense of inquiry that brought about his downfall. We are too clean cut. I think the Greek mind was much more sophisticated in that it realized that it could be both and you did not need to choose one or the other. This is certainly how it is presented in the tragedy.
On the one hand, it was prophesied that Oedipus would kill his father and marry his mother. So, from this perspective, it was fate pure and simple. To prevent this, he was send away as a baby, presumably to be exposed, but he lived.
On the other hand, Oedipus has this insatiable desire to find out solutions and save his city, Thebes from a plague. Hence, he drives the play with his questioning. Even when others urge moderation, he continues, until he discovers the horrible truth of what he did.
In the end, we can say that it is fate and Oedipus' sense of inquiry that led to his downfall. Life is filled with mysteries and this is one of them.