If so, when were they “adopted” into white society? What are some other examples of something similar occurring in other areas of the world? Do you see this happening again with other ethnic groups in the US as the percentage of white Americans decreases?
The relative "whiteness" of various European immigrant groups is one of the most common questions I've gotten as an immigration historian, and in my opinion it is one that we as a field have done a terrible job at adequately answering. I am still probably going to fail at that task in this short response, but bear with me.
"Whiteness Studies," particularly studies that examined the relationship between whiteness and European immigrants, saw a flurry of activity in the 1990s, anchored by works such as David Roediger's The Wages of Whiteness (1991) and Matthew Jacobson's Whiteness of a Different Color (1999). These works sought to do several things, chief among them figure out who counted as "white" in different cultural, legal, and (pseudo)scientific contexts. The top-line, undeniable takeaway of this body of scholarship is that throughout Western history, the definition of who "counts as white" has been unstable and evolving. Whiteness, therefore, is not simply a matter of pigmentation, but a reflection of various other social factors.
We run into trouble, however, when we try to disentangle the various factors that made someone "white" or "not white," and figuring our which ones mattered the most. The work I've most commonly seen cited in discussions of Irish racial ambiguity is Noel Ignatiev's How the Irish Became White. This is obviously an extremely provocative title, and it is one that I find frustratingly misleading for reasons that I will get into below. Ignatiev's guiding question is how the Irish moved "from oppressed to oppressing," (1) and, drawing heavily from Roediger, focuses on post-Civil War developments in labor that drew a social distinction between "black work" and "white work" (111-112). The arc, according to Ignatiev, goes from the treatment of the Irish by English colonists in Ireland (what he calls "a classic case of racial oppression" (35)), to a "white republic" that emerged from the ashes of both slavery and nativism. While he raises several important points in evaluating how the Irish achieved the economic and personal freedom, as well as "citizenship" (which he leaves poorly defined) that he considers to define 19th century whiteness (2-3), I have several problems with Ignatiev's argument.
The first and most damaging is that the United States government has always legally considered the Irish to be white.* Congress has the enumerated responsibility for setting naturalization policy, and passed the Nationality Act of 1790, which limited the right to naturalize to "any Alien being a free white person." This law remained in place until 1870, when naturalization was also extended to "aliens of African nativity and to persons of African descent." This was not modified again until 1906. Throughout this entire period, there was never any serious challenge to the common sense understanding that the Irish were white for the purposes of citizenship law. To be sure, there were challenges to the voting rights of naturalized Irish-American citizens in some states during the Know-Nothing movement of the 1850s. The most well known of these was the 1858 Two Years' Amendment in Massachusetts, which denied the right of suffrage to naturalized immigrants until two years after they obtained citizenship. However, while this sprung from widespread anti-Irish hostility among "Old Stock" Massachusetts residents, the impetus for this hostility was much more anti-Catholic sentiment than a sense that the Irish were not "white."
Second, the arguments of Ignatiev and others who focus on anti-Irish discrimination draw disproportionately upon episodes from the eastern seaboard in general and from Massachusetts in particular. For example, Hidetaka Hirota's excellent Expelling the Poor: Atlantic Seaboard States and the Nineteenth-Century Origins of American Immigration Policy convincingly demonstrates that anti-Irish animus was strong enough in the 19th century to bring about some of the first organized deportations of paupers and the disabled. This would suggest that anti-Irish sentiment has been foundational in American attitudes towards who does or does not count as a potential American.** These efforts to disenfranchise and to culturally or economically marginalize the Irish were very real. At the exact same moment, however, Irish-Americans in other parts of the country were being empowered through their legal identity as white to seize greater economic and cultural control. This was most visibly on display in the far west, where Irish workers were key in the anti-Chinese movements in California and elsewhere, and evident in slightly less dramatic form in states such as Wisconsin, where white immigrants were given suffrage rights even before formally obtaining citizenship in the interests of expanding the electorate and expediting statehood.
My third point is more speculative, but I get a sense that for as ground-breaking as Jacobson's Whiteness of a Different Color is, it may be somewhat misused. The late nineteenth and early twentieth century shift towards "scientific racism," as Jacobson clearly demonstrates, meant that people were completely in earnest when they divided whites into racially distinct Teutons or Anglo-Saxons or Slavs or Mediterraneans. The former two were culturally, economically, and pseudoscientifically white, while the latter two were not, but were legally classified as white for the purposes of immigration law (though the government sought to limit their access to the US through the Johnson-Reed Act of 1924). I suspect that much of the conversation surrounding the Irish specifically reads that view of the world anachronistically into the early 19th century.
In sum, legally speaking, the Irish have always been white. Culturally and economically speaking, it gets more complicated. The task, then, is to figure out if discrimination being directed against the Irish on cultural and economic grounds takes away their "whiteness." It's fair for people to land on either side of that debate, although my personal feeling is no, given that:
the most intense period of anti-Irish discrimination preceded the heyday of scientific racism. I think it is entirely legitimate to talk about how Italians or Jewish immigrants became white (as Matthew Guglielmo and Eric Goldstein do), because the largest periods of their immigration coincided with pseudoscientific arguments that expressly and predominantly couched opposition to their immigration in racial terms. At this point, however, the Irish had been explicitly cast as part of the "Old Stock" of American society, alongside Anglo- and German-Americans (this, by the way, would be the answer of when they were "folded into" a more or less unimpeachable sense of white identity as we now think of it).
it relies on a "common sense" linkage of whiteness and citizenship in all its legal and cultural forms. Ignatiev relies (I suspect without realizing it) on a post-Reconstruction conception of citizenship as a binary: either you're a fully equal member of the body politic or you're not. In reality, questions of citizenship and belonging were much more nuanced during the main period of Irish immigration. Some scholars who have touched upon this include William Novak and (through studies of African Americans) Martha Jones and Stephen Kantrowitz. The argument from Ignatiev is that groups who suffer discrimination are ipso facto not white and can only truly establish their whiteness through labor and voting rights, which I find a dubious assertion.
At the end of the day, the answer to legal whiteness is clear. Cultural and economic whiteness gets more tricky. Do you need to have, as Ignatiev puts it, the right to be an oppressor rather than oppressed? Does your religion need to be considered mainstream? Do laws directed against your group, such as prohibition, mean you're a population that needs to be governed, and therefore bear the markers of non-whiteness? Are cartoonish caricatures enough to designate you a racial other, and if so, how grotesque do the caricatures need to be? I don't think we have any good or clear answers to these questions, and certainly no consensus ones, but that's what the conversation hinges on.
*I am not touching here upon the claims made by earlier English authors that the Irish in Ireland were racially distinct, just the Irish-American experience
**My own work suggests that Hirota captures something big, but misses a key dynamic that leaves us with a disproportionate emphasis on "Old Stock" fears and actions in the origin of immigration policy