Ken, as a descendant of Italian immigrants, I think you should know that many Americans at the time were like you: idiotically fearful that Italian immigrants would change the culture of the United States and were not equals to "whiter" Americans already here. 

By the late 19th century, the peninsula of Italy had finally been brought under one flag, "but the land and the people were by no means unified. Decades of internal strife had left a legacy of violence, social chaos, and widespread poverty," the Library of Congress notes. "The peasants in the *primarily poor, mostly rural south of Italy and on the island of Sicily had little hope of improving their lot. Diseases and natural disasters swept through the new nation, but its fledgling government was in no condition to bring aid to the people.* 

"As transatlantic transportation became more affordable, and as word of American prosperity came via returning immigrants and U.S. recruiters, Italians found it increasingly difficult to resist the call of “L’America”.

"This new generation of Italian immigrants was distinctly different in makeup from those that had come before. No longer did the immigrant population consist mostly of Northern Italian artisans and shopkeepers seeking a new market in which to ply their trades. Instead, the vast majority were farmers and laborers looking for a steady source of work—any work. There were a significant number of single men among these immigrants, and many came only to stay a short time. Within five years, between 30 and 50 percent of this generation of immigrants would return home to Italy, where they were known as ritornati."

Italians looked different than the Northern European immigrants that were whiter in skin tone. They weren't turned away. And you shouldn't be turning away non-white immigrants that aren't from Europe - because had that policy been in place when your ancestors sought a better life in the United States - you wouldn't be here ruining the American dream with your ignorance and racism.