Oedipus' hubris and insatiable quest for the truth paradoxically lead him to blind himself for having known the truth all along.
Oedipus is one of those plays, like the great Shakespeare tragedies, where the audience knows what's going to happen to Oedipus before the play starts. The story was well-known then, as it is today. Yet, it still elicits catharsis (pity and fear) from us all because we identify with his quest for the truth and we dread digging up skeletons in our own family closets.
Yes, it could be fate or free will that leads him to his blinding, but that's a blanket statement that covers every possible outcome in between. Regardless of what happened to Oedipus when he was a child (his crippling), regardless of the Oracles, regardless of the Riddle of the Sphinx and the plague that besets the city, Oedipus can still choose not to know. Tieresius warns him not to pursue his questioning of the shepherd. Oedipus has been so smart and wise, so he thinks, in securing the crown that he arrogantly thinks he can ferret any truth left unknown.
Fate and free will aside, Oedipus had an "out:" he could have lived the rest of his life, albeit plagued, but humbly not knowing his family's gruesome past. He chose to know, and though other, weaker men (or women like Jocasta) might have killed themselves knowing such atrocities, Oedipus went on to have a sacred, blessed death.
In so many ways, this is a fairly powerful question that strikes at the heart or essence of the debate over fate and free will. On one hand, I would suggest that Oedipus' own actions constitute a major reason for his downfall. I would say that Oedipus' inability to heed to words of the prophecy about his own fate as well as his belief in his own sense of infallibility contributed to his downfall. His own sense of hubris, excessive pride, played a very large role in the events that befall him. At the same time, I would suggest that fate plays an equally dominant role the drama of Oedipus. I think that fate underscores so much of what happens to him that it is a web that ensnares Oedipus. It was almost as if that Oedipus tried in vain to outrun his fate.