There will be has been and will be more arguments and debates over who is "White" or not.
Consider that when the United States was founded (a generally WASP project) this was the first seal that was ultimately rejected and was meant to represent the "Countries from which these States have been peopled" which consisted of England, Scotland, France, Ireland, Germany, and the Low Countries (Netherlands):
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It sometimes sees use today as symbol for Old Stock Americans which represents a demographic of Americans who are descended original settlers of the Thirteen Colonies before the waves of Italians, Scandinavians and Polish.
Even in the early 20th century this was an issue for the United States considering that
United States v. Bhagat Singh Thind happened which amounted to an an Indian Sikh man who identified himself as an Aryan trying to be naturalized as a citizen on the basis that he be classified as a "free white person" within the meaning of the Naturalization Act based on the fact that Indians and Europeans share common descent from Proto-Indo-Europeans. Thind ultimately lost the case on the basis that his argument did not meet the common sense definition of what a White person is.