Yep. In fact, many ethnic groups from Europe have become “White” over time - Slavs, Italians, Greeks, and Eastern European Jews. Typically because they a) came to America in sizable numbers (escaping famine or war at home) b) struggled to “assimilate” (immigrants tended to be poor or working class, spoke little English, maintained many of the “old ways” regarding dress, food, and so forth, and c) were phenotypically different from Western/Northern European immigrants that came generations before (the “swarthy” European classes).
As a result of discrimination and pluck, they tended to settle in ethnic enclaves. Which explains why there are Little Italys and Greektowns in many East Coast urban cities. While many did in fact assimilate, learned English, dropped the foreign-sounding names… many others had accents that were too thick, or were too immersed in their native culture and refused to surrender it (despite the paying a social and economic price for doing so).
But for most immigrants, the third generation is where the Americanization process was complete. There’s a great book by Karen Brodkin called “How the Jews Became White Folks,” which goes into this issue at great depth.