Rosie has been a longtime friend of Frank Murphy, though their friendship began to deteriorate after he saw Frank insult African-American people on TV, and after Frank befriended the Airlines' CEO, Mr. Dunbarton.
Rosie's friendship with Frank was eventually restored after Frank resolved the conflict that led to a strike. He also helped Frank receive new employment while Frank helped Rosie and the other airline employees deal with their new boss, Scoop Dunbarton.
He also has an unnamed daughter who Frank is the Godfather of.
Rosie has to deal with racism often, especially at work. This is a major theme for him in Season 3.
Rosie has an illegitimate, puerto-rican son named Rosie Jr who he when he was 17 at a puerto-rican day parade. His relationship with his son seems to be somewhat complicated as he prefers not to talk about him and his were not even aware of the fact that Rosie Jr. was their half-brother, believing he was their uncle.