Is one sinful if he does not speak to his neighbor after an argument between them, even if there was not much communication between them to begin with, and the period of not speaking exceeds three days? And is it necessary to apologize to him if the neighbor avoids the one who quarreled with him?

Allah has enjoined good treatment of neighbors in the Quran and Sunnah, and there are hadiths that encourage showing kindness to neighbors and warn against harming them. You are obliged to treat your neighbor well and not cease greeting him for more than three days, even if you feel he is avoiding you. You must strive to mend relations between you and him, and apologize to him, for that is better and brings a greater reward.