Dominick's grandfather and namesake acts like he's the second coming of Christ. His religious fervor makes him seem just as crazy as Thomas, except in his case, crazy seems to look like religious fervor. He (allegedly) sees a statue of the Virgin Mary crying, and takes this to be a sign from God that he is a chosen one.
He's paranoid (another parallel with Thomas), believing that people are out to get him under the premise that "jealousy of superior men" (33.33)—like Domenico—is "everywhere" (33.33). And he's cocky (like both Dominic and Thomas) to boot, calling himself the "most gifted student" with "considerable native talents" (31.17). You know, if he does say so himself.
Like Dominick feels he has to take care of Thomas, Domenico feels responsible for his brothers. His brothers, Vincenzo and Pasquale, don't last long after they relocate to America. Vincenzo can't keep little Italy in his pants and gets caught with a cop's wife by the cop himself, who shoots Vincenzo right in his Apennine Peninsula, which gets infected and kills him. And Pasquale falls from a ladder to his death, too.
Domenico eventually buys himself a wife who has two children—one whom dies at birth, and the other whom eventually becomes Dominick's mother.
Dominick hopes to find out who his real father is by reading his grandpa's memoir. He suspects his dad is Angelo Nardi, the man who typed Domenico's memoir for him. But Angelo Nardi must be Italian for red herring, because he isn't Dominick's dad, and we never even meet Nardi during the course of the novel.
What Dominick learns is that his grandfather is a cheapskate, an angry man, and a monkey killer. And by the time Dominick comes into possession of the memoir, he's a monkey and a burlap sack full of bricks away from falling in line with grandpa's footsteps. After reading the memoir, Dominick thinks, "Don't be him, Dominick" (46.9), using Domenico as an example of how not to live life.