In order to avoid unwarranted assumptions, by "becoming white" you really mean how Italians came to be accepted into mainstream American society, right? I am asking because the wording of your question is somewhat ambiguous.

Well, anti-Italianism in Britain is one of my side-tracked areas of specialty, but British and American society are pretty in tune with each other on this topic at least, let's see what I can tell you about anti-Italianism in America. Some salient points:

1. Anti-Catholicism and anti-Italianism are very, very linked. Decide when you think the majority of Americans stopped seeing Catholics as "other" and you will get an easy pulsepoint on anti-Italianism and anti-Irish sentiments both. The election of JFK is often used as a timepoint for when Americans were majority accepting of Catholics.
2. The third generation after immigration is usually the first generation to lose their native language and speak only the new language. For the bulk of Italian-American immigrants coming to American in the late 19th and early 20th century, the third generation would be born in the 40s-50s. These children are "fully American" and would come of age in the 50s-60s, a decently tidy line-up with our JFK date from above.
3. I've seen the crazy stardom of Rudolph Valentino listed as an influence in making White America see Italians as attractive, sexually desirable people and not criminals, but I find it a weak argument as that was in the 1920s and Italians were still very "other", but other famous and liked Italians working in the 30s-50s like Joe DiMaggio, Dino Martin, Frank Sinatra, etc. should not be overlooked.
4. The Sicilian Mafia's loss of influence after prohibition certainly helped a heck of a lot, having Italians in the newspaper all the time for crime is not good for your application to White American.

I've cribbed these notes together a few minutes before leaving work simply because I was annoyed at all the crappy arguing in this thread and you deserved an answer, so let me know if anything's too sloppy to understand. I can also show you some interesting stuff about who runs the Met Opera and how it moved from Italian managers to Americans around this time frame, that's pretty interesting.

This might be a better question for r/asksocialscience.

You could also read Working Towards Whiteness: How America's Immigrants Became White. I've not read it, though I've seen it around, seen it cited, etc. Your professor was most likely referring to this idea when she made this reference. TL;DR from Amazon's description of this book is that, according to the author, it was a combination of the labor movement, New Deal, post-WW2 social changes, and other 20th century developments.

hi! you'll find some info and links to several previous posts here

• How did the Irish, Italians, and Jews become "white" in the United States?

This may not be an acceptable answer, but initially, "white" appears to be the designation applied to Americans of English/WASP descent, as according to the US Census "racial" designations, now included as "white" are Germans, Irish, Spanish, Italian Greek, North African, Middle Eastern (Jordanian/Iraqi/Syrian/Iranian/Lebanese/Palestinean/Israeli), Religiously, Jews were finally seen as "white" in the 1970's

"Whiting" an immigrant group could have been seen as weakening any type of alliance with African-Americans