I think the phrase used in GFIII is another example of poor plotting. There has never been an out-and-out "war" between American and Sicilian Mafias because the two have operated exclusively on their own turfs. If anything, drugs have united them--the Sicilians have organized the manufacture and shipping of heroin and morphine to America, the Americans have been in charge of distributing it here.
The closest to a "war" was a skirmish in the Bonanno Family in the 1970's and '80's. Sicilians (called "zips") were brought into the family to help with the drug traffic (the famous "pizza connection"). The "zips" attained some power in the Bonanannos, but they were defeated in the war that followed Carmine Galante's assassination in 1979, leading to Rusty Rastelli's elevation to Don.