Eddie Carbone is deeply conflicted and confused. He is unable to be honest about his feelings for his niece, though others can see clearly that Eddie's fatherly affections for Catherine have at some point morphed into a romantic and jealous love.
...Eddie has unarticulated and perhaps unconscious desires to possess Catherine himself.
Beatrice attempts to persuade Eddie to let go of Catherine early on, but cannot directly confront Eddie with her views of the situation. This oppression of the clear truth is a theme of the play. Open secrets are at the heart of this play's dramatic conflict.
Eddie's love for Catherine is clear yet remains undisclosed and un-discussed. This is paralleled by Marco and Rodolpho living in the neighborhood as illegal immigrants. Everyone knows that the two are illegally working, but nothing is ever said openly in public about it.
Another theme expressed through Eddie (and Marco) relates to a certain tortured confusion of the role of the protector. Eddie and Marco are both patriarchs who feel they must protect their families. For Eddie, this protective urge becomes a misplaced romantic affection. Eddie unwittingly merges his role as caretaker and lover, an emotional act that causes him to be jealous of Rodolpho's interest in Catherine.
Later, Marco finds himself called upon to protect Rodolpho in a way that confuses protection with revenge. This confusion of the protective urge leads to violence at almost every turn in the play. Propelled by what they feel is an emotional necessity, Eddie and Marco both demonstrate the ways in which role and identity can be as unconscious as they are conscious, leading to out of control behavior.
When deeply felt passions are in play and out of control, horrible things will emerge.
This is not merely a case of masculinity gone awry. Eddie and Marco are driven by the cultures they have internalized. The two men express a conflict that is much larger than two individuals.
Ultimately, the majority of Eddie's character traits are expressive of his inability to know himself and his tragedy is born rather directly out of this flaw.