Pub. L. 82-414 § 301(a)(7), Pub. L. 82-414 § 301(b)
Cruz is defined in law, “citizen”.
Yes, Mr Cruz was a citizen of the US at his birth and thus was naturally born as a US citizen. He never was an “alien” and never was in need of naturalization and he never immigrated.
I would also point out that in Title 8 section 1401 section A, anyone born in the US is defined by law as “citizen”.