It is a very small gesture but symbolises a lot. He stands up to get some sort of physical presence, he has acknowledged what has just happened and is reacting to the situation, and you could say that he rises to the occasion. Rodolpho has just started to understand the full situation, with a certain gleam, he smiles back at Eddie, and by this simple gestus he is retaliating to Eddie hit. He has now started to fight back. Beatrice has to physically pull Eddie back into his chair to try and stop him and to calm the situation down.
Eddie reacts to Rodolphos smile by offering to teach him again. By doing so it will give him another chance to hit Rodolpho again, although on another level he is implying that he will teach Rodolpho a lesson for taking Catherine away from him. Rodolpho knows he has won this battle and to rub it into Eddies face, he asks Catherine to dance. Dancing can be seen as quite a sexual thing, so it would be improper of Eddie to start dancing around with his niece.
So by Rodolpho dancing with her, he is claming her, making a statement. It is now Marcos chance to make his statement and challenge Eddie, his actions are telling everyone that he is now in charge, that he is strong and that if Eddie wants to mess with Rodolpho, he is going to end up messing with him too. He does this by challenge Eddie to lift a chair, knowing that Eddie wont be able to; he is also trying to take the attention off of Catherine and Rodolpho.
Eddie falls to lift the chair, Marco then lifts it. This way he is showing everyone that he is the strongest of them all. Marco never made this statement before, as he never had the need to, and he respected Eddie, but now Marco feels the need to prove to him that he is more macismo. By the end of the scene we can see a define barrier between Eddie and everyone else, with Beatrice in between. Rodolpho and Marco understand why Eddie is angry and Catherine is angry with him as he hit Rodolpho.
Within the scene there is an element of foreshadowing what has been shown here will be shown again later on in the play. This is true as the barrier stays up until the end, later on when Rodolpho and Catherine are going to get married there is still competition between Marco and Eddie, because of the situation with Rodolpho and Catherine, and Beatrice is left trying to calm everyone down, stuck in the middle as before.
There is no resolution to this argument, it is now between Marco and Eddie and neither of them will back down as they both believe that they are in the right and neither of them want to be shamed or brought down to a lower level. Thats why, unfortunately, there is only going to be one solution, and in this case it is where Eddie dies, this can only be blamed on Italian heritage.