Suppose that a person was born in a independent state and later (i.e. after the birth) this state was admitted into the United States. Would such a person be considered a natural born citizen of the United States (which is a requirement prescribed by the US Constitution for holding the office of the president or vice president)?
It is slightly complicated
No such case has occurred in US history, so no case law is fully on-point.
It should be noted that in such a case it is quite likely that the parents of such a person would be quite likely to be US citizens, in which case such a person would, under current law, be a citizen of the US from birth (if a parent had resided within the US for at least a minimum time.
In [United States v. Wong Kim Ark, 169 U.S. 649 (1898) (https://supreme.justia.com/cases/federal/us/169/649/) it was said at 169 U. S. 715:
In my judgment, the children of our citizens born abroad were always natural-born citizens from the standpoint of this Government.
This is dictum, as it was mote directly relevant to the decision.
earlier, at at 169 U. S. 715 it is said:
[C]itizenship by birth is established by the mere fact of birth under the circumstances defined in the Constitution. Every person born in the United States, and subject to the jurisdiction thereof, becomes at once a citizen of the United States, and needs no naturalization.
This is binding precedent.
Note that this applies to any part of the US, including territories that have not yet become states.
According to Vox in "The “natural-born citizen” ceiling":
The Supreme Court has stated that, properly understood, the definition of “natural-born” covers anybody who was a US citizen at birth, meaning they did not have to go through naturalization at a later time. In the 1898 case United States v. Wong Kim Ark, the Court explained that, in British common law, “natural-born British subject” meant “a British subject who has become a British subject at the moment of his birth. […] Any person who (whatever the nationality of his parents) is born within the British dominions is a natural-born British subject.”
- A person who is born, outside the US, to US parents (or one US parent) under the proper conditions so that such person is a citizen at birth, is eligible to be elected President of the US.
- A person who is born within an incorporated territory of the US is a natural-born citizen, and is eligible to be elected President of the US, whether that territory later becomes part of the US or not.
- A person who is born to non-US parents in a plavce that is then part of a country other than teh US is not a natural-born citizen, even if that place latte becomes part of the US and is admitted as a state.
The only relevant example using the US Constitutional naturally-born requirement is Barry Goldwater, who was born in Arizona Territory. The lack of a legal challenge to his candidacy is not definitive proof – this is a matter that the Supreme Court has not ruled on. Various others were citizens because of parental citizenship (e.g. Ted Cruz and John McCain). Charles Curtis, 31st Vice-President, was born in Kansas Territory, and was a member of the Kaw (Kansa) Nation. Also, Al Gore was born in Washington DC, which is still not a state. Gore's parents were US citizens, I don't know the facts for Curtis' parents. Complicating the matter is that the question asks about an "independent state" (nation, I assume) – Arizona and Kansas were US Territories. If Iceland were to become a US state, it's legally undecided whether a native born Icelander would become a "natural born citizen".