Like every presidential election, candidate qualifications will be determined by the voters and their electors. The Constitutional standard is that a candidate be 35 years old, a resident of the US for 14 years and be a natural born citizen. The Constitution requires that the president be selected by the voters and a majority of their electors. Both voters and their electors are obliged to vote only for eligible candidates and they must determine eligibility as part of their constitutional function to select the president.
So, if there are people who believe that Cruz is not eligible for any reason, those people should make their pitch to the voters and their electors. If Cruz is then selected by a majority of the electors, that then is the end of the matter.
All else is baloney.
Your response is consistent with the judicial doctrine of “political question.”
POLITICAL QUESTION DOCTRINE
Federal courts will refuse to hear a case if they find it presents a political question. This phrase is construed narrowly, and it does not stop courts from hearing cases about controversial issues like abortion, or politically important topics like campaign finance. Rather, the Supreme Court has held that federal courts should not hear cases which deal directly with issues that Constitution makes the sole responsibility of the other branches of government. Baker v Carr, 369 U.S. 186 (1962). Therefore, the Court has held that the conduct of foreign relations is the sole responsibility of the executive branch, and cases challenging the way the executive is using that power present political questions. Oetjen v. Central Leather Co., 246 U.S. 297 (1918). Similarly, the Court has held that lawsuits challenging congress’ procedure for impeachment proceedings present political questions. Nixon v. United States, 506 U.S. 224 (1993).
Eligibility is determined by law, not plebiscite.