Could Oedipus have prevented the prophecies from coming true?
Of course, since the story of Oedipus is a play, which is a work of fiction, anything could possibly happen. But it is important to note that both Oedipus and Jocasta attempt to take matters into their own hands and defy the prophecies. Oedipus learns, while living with the man he believes is his father (Polybus), that there is a prophesy that he will kill his father and marry his mother. He decides to leave and never return in order to avoid the prophecy's fulfillment.
It is upon this decision that he actually moves, unbeknownst to him, right into the eye of the storm. He meets his true father, Laius, on the road, and since both are too stubborn to give way to the other, they reach a stalemate, which Oedipus solves by killing Laius. He then solves the riddle of the Sphinx which wins him the hand of Laius' widow in Thebes, Jocasta, who is his mother. So the prophecy is fulfilled as a result of Oedipus attempting to avoid it.
Jocasta contributes to this when she gives Oedipus as a baby to a shepherd to "dispose" of because of the prophecies that she learns about.
For the Greek audience, the Chorus of this play is teaching the important lesson that humans are foolhardy and disrespectful to imagine that they might outsmart the will of the gods. This implies that the prophecy is the will of the gods, and, as such, is to be endured rather than avoided.
I do not believe that Oedipus could have prevented the prophecies from coming true. First, the people of the time when the story originated believed in prophecies, usually presented by oracles. They believed in their veracity.
Second, even as his parents abandon him to prevent the prophecy from coming true, it is this abandonment that lead Oedipus to grow up in a different household, not knowing that his adoptive parents are not his real parents. In leaving his adoptive family in trying to thwart the fate the prophecy has set in place for him, he is actually moving toward that fate unknowingly.
Oedipus does, in fact, kill his father, without knowing who the man is. He is welcomed into Thebes (his actual home town) as a hero. (He has made the roads safe by defeating the murderous Sphinx.) As a hero, his mother is offered in marriage to him, though she does not recognize her son.
This culture believed in pre-destination: that once your fate had been decided, you could more easily defeat the gods (impossible) than flee from your destiny.