According to Aristotle, a tragic hero is defined in part as a person who has a tragic flaw which precipitates his or her own downfall. Eddie Carbone's tragic flaw is his jealousy. He is jealous of his niece, Catherine, and this jealousy gradually destroys his relationship with his wife and eventually leads to his downfall when Catherine falls in love with another man.
When Eddie comes home one day to see Catherine and the other man, Rodolpho, emerge from the bedroom, he loses all self-control. He grabs Catherine, his niece, and kisses her forcefully on the mouth, and then he threatens Rodolpho. This is the beginning of Eddie's demise. He contacts the immigration authorities about Rodolpho, who is in the country illegally, and in doing so, he commits an act of betrayal which will ultimately see him despised and ostracized by his community.
Eddie Carbone is also a tragic hero because he is pitiful. He is, in many respects, a good man. He works hard and provides for his family. He loves his wife and has been a good father figure to his niece. He takes Rodolpho and Marco into his own home when they have nowhere else to stay, and he looks after them, at least until he discovers that Rodolpho has feelings for Catherine. When he dies in his wife's arms at the end, he is a tragically pitiful figure.