The Naturalization Act of 1790 includes "white" as a legal category, quoting, "That any Alien being a free white person..." at that time, and in subsequent acts explicitly the "free white person" eligibility language included European immigrants.

The idea that "whiteness" is a mere sociological category is ahistorical invention of what was a fringe part of sociology, now "whiteness studies", and its mostly "not even pseudoscience". These "studies" speak passively of "what whiteness is" as if sociologists of this type are in some sort of telepathic mental conversation with the 18th century.

Foruntely, this particularly claim about irishness and whiteness has a factual interpretation, so its trivial to show it is false. The european diaspora into america was always "white", and this is trivial to show in 1790 let alone afterwards. "White" refers to the family of european ancestries which 'look white' and this is written into the text of early american law, let alone the ordinary meaning of the word

EDIT, since this is downvoted, here you go:

Irish naturalised throughout the late 18th, 19th centuries explicitly under this 1790 law which limited naturalization to "free white persons". QED. There's no actual debate here. This "irish werent white" idea is a mid-90s invention.

See, eg., at the heights of alleged anti-irish sentiment,

> United States v. Bhagat Singh Thind, 261 U.S. 204 (1923): ... The words of familiar speech, which were used by the original framers of the law, were intended to include only the type of man whom they knew as white. The immigration of that day was almost exclusively from the British Isles and Northwestern Europe, whence they and their forebears had come. When they extended the privilege of American citizenship to ‘any alien being a free white person,’ it was these immigrants — bone of their bone and flesh of their flesh — and their kind whom they must have had affirmatively in mind. The succeeding years brought immigrants from Eastern, Southern and Middle Europe, among them the Slavs and the dark-eyed, swarthy people of Alpine and Mediterranean stock, and these were received as unquestionably akin to those already here and readily amalgamated with them.