Your question hits the nail right on the head. One of the reasons why Oedipus Rex is a timeless classic is owing to the fact that this question cannot be answered easily.
On the one hand, it is fated that Oedipus would marry his mother and kill his father. To put it another way, if there is a prophecy and the will of the gods is infallible, can a man avoid it? Oedipus thought he could. When he heard of this prophecy, he did something noble, he left his city and his family, so that the prophecy would not be fulfilled. From this perspective, we can say that he is not responsible.
On the other hand, Oedipus was a person of hubris, that is, pride. He thought he could flee and outwit the prophecy. Also he had a violent streak in him. He struck a man who almost ran him off the road. Unknowingly, he killed his father by that act. Also when he became king of Thebes, he thought he could solve every problem. By seeking an answer to the plague of Thebes, he began to unravel the mystery of his past and identity. So, from this perspective, he is responsible.
In the end, we are in a dilemma. It is the old question of fate and free will.