I'm not sure about voting per se, but in terms of immigrant integration and policy more broadly, it's useful to remember that not all European immigrants in the 18th and 19th Centuries were treated as "Europeans" equally -- that is to say, eugenics-like "theories" and thinking in policymaking circles led to the passage of laws like the National Origin Quota Act (1924) that ostensibly eschewed race-based quotas in favor of preserving the "status quo" of lineages present in the U.S. at the time; in other words, it resulted in a "freezing of society" so that immigrants from, say, North/Western European countries would be admitted at rates reflective of the proportion of the existing U.S. population that could trace its lineage to those afore-mentioned regions/countries. As immigration from those particular regions of Europe had outpaced flows from Southern and Eastern Europe up until around the late 1800s -- to say nothing of immigrants from East Asia, who were actively excluded via U.S. immigration policy -- this essentially ensured some sense of "racial" homogeneity couched in the more liberal (and fashionable, I might add) language of "national origin," rather than "ethnicity" or "race" of immigrants. A good source on the "de/re-ethnicization" of immigration policy in the US and other immigrant-receiving countries is Christian Joppke's Selecting by Origin: Ethnic Migration in the Liberal State (2005).