There are many elements of this play unrelated to immigration. In fact, the central drama is not directly related to the circumstances of Marco and Rodolpho as illegal immigrants.
Rather, the central conflict of the play and its dramatic impact is drawn from Eddie Carbone, his jealous and innappropriate feelings for his neice, and his inability to understand himself. The rivalry between Eddie and Rodolpho grows out of these elements.
Eddie’s desire for his niece is exacerbated by her growing affection for her second cousin.
An audience will recognize this scenario as a universal one, without constraint or contingency on the issue of immigration. This is a tragic scenario, wherein family affections lead directly to role-based dramatic conflict and inevitable confrontation.
The flawed father-figure brings suffering to his family and though his flaw leads him to attempt to punish others and inflict harm, the greatest harm is ultimately done to him. Eddie's inability to understand his feelings for his niece and to control his passions prove to be his undoing.
Miller based this play, formally, on Greek tragedy, a medium meant to be expressive of large and even universal themes. In his introduction to the play, Miller writes at length about his interest in creating "social drama", a mode defined by its aims for communicating ideas relevant to society as a whole.
[Miller] decided to create an American myth using many of the structural devices found in the works of the Greek masters.
This intention speaks to the secondary nature of immigration in the play. Additionally, we can read the immigration element as symbolic. The Italians, Marco and Rodolpho, represent an "old world" that exists in a complicated imbalance with the new world (both in contemporary theater and in American values regarding otherness).
Audiences do not need to be versed in immigration history or politics to relate to the conflicts and themes of the play, which attempts to be classic and universal in its approach to drama.