Before I attempt to answer the question, it is important to know something of the Greek worldview. At the center of the Greek world, there were two maxims at the Oracle of Delphi: "Know Thyself" and "Nothing in excess." These two points expressed the importance of keeping human boundaries and not transgressing into the realm of the gods. From this angle, we can say that the Greek were humble. In other words, they should know that they are not divine.
Against this backdrop the tragic flaw of Oedipus was his thought that he could solve any problem. He thought of himself as more than human. So, when at the beginning of the play there was a plague, he thought he could avert it. Rather than beseeching the gods or even asking for help from others, he sought to rectify any problem alone. The Greeks called his quality, "hubris." In short, we can say that pride got the best of Oedipus. This was his tragic flaw.