The gods were disgusted that Oedipus, having killed his natural father, was not exiled or killed due to bloodguilt, and strike a plague on the city.
I do not agree with his moment of realization as the point where catharsis is reached. Indeed, as demonstrated in Oedipus Rex, while there is free will in the human condition, there is also predestination to one's life.
There are other ways to look at the case. Oedipus traveled to Thebes and answered the riddle correctly.
Hedda Gabler and Oedipus both find a way of relieving themselves through self-punishment, simply because it is the only way of overcoming their obstacles The citizens of Thebes consider him a hero because he restored harmony to their kingdom. Also if there were no characters the outcome of the play would be different I think of a very ignorant man that tries to escape his fate-- a man that had so much confidence in himself that he would put false accusations on people and defy his gods just because he thinks he is right.