According to Aristotle, a tragic hero has several key characteristics.
1. A flaw or error of judgement
2. A reversal of fortune brought about because of the error in judgment
3. The hero should recognize that their fortunes changed because of their decision
4. Excessive pride
5. Hero's fate must be greater than deserved
Some lists add that the hero must be of noble birth. While true of characters like Oedipus, King Lear, and Brutus, this does not apply to Ahab. Readers know that he was orphaned, but little else. He could be considered nobility though based on him being captain of a ship.
His fatal flaw is his supreme overconfidence. He believes that through sheer will and determination he can bend nature to his will and defeat the white whale. That's a very prideful opinion of one's self.
Lastly, Ahab's eventual downfall is brought about because of his inability to give up, and his fate is greater than deserved because he is not a villain. The reader doesn't wish him dead and/or a horrible fate to fall upon him.