Cruz acquired citizenship via federal statute granting citizenship to individuals born outside the U.S. to U.S. citizen parents (Pub.L. 82–414 § 301(a)(7); 66 Stat. 236).
To retain citizenship he was required to fulfill the requirements of Pub.L. 82–414 § 301(b); 66 Stat. 236.
The retention requirements of § 301(b) where eliminated in 1978 (Pub. L. 95–432 § 1,3; 92 Stat. 1046). His citizenship is via Congressional grant, i.e. he is naturalized.
The Supreme Court is clear on this point: statutes “conferring citizenship upon foreign-born children of citizens” is naturalization (see U.S. v. Wong Kim Ark, 169 U.S. 649)
A natural born citizen does not need to take affirmative actions to retain citizenship nor can their citizenship be revoked, to the contrary, a natural born citizen must take affirmative actions to renounce citizenship. (see Afroyim v. Rusk, 387 U.S. 253).
Well, I do not think that there is anything that the Supreme Court can do to resolve this issue. I believe that the Constitution delegates to Electors the job of choosing the President and also provides the Electors with some standards regarding eligibility (age, natural born citizen, residency). I believe that it is the function of the Electors to choose a President who is qualified to serve. They are provided with standards to apply and entrusted with the duty to apply those standards. I do not believe that their decisions are reviewable by the Supreme Court, by either branch of the legislature, or by the President. In the event that a majority of the Electors cannot agree on a President, then the House of Representatives is to choose the President, voting by state.