DY……..
Who gets on a ballot in an individual state is a state law question, not a FEDERAL question.
The primary elections are state elections, not federal and the federal courts cannot impose their rules on such. That would violate the states rights guaranteed under the 10th Amendment to the US Constitution.
The States determine who can be on the ballot and who can vote.
In some states convicted felons never get to vote. Here in CT after a convicted felon has completed his/her prison time, probation and/or parole and made all ordered fine and/or restitution payments the vote may be restored. Some states never give the vote back, and a few let felons vote from prison.
In many states you must be a registered member of the political party to vote in its primary election (CT requires this). Other states have crossover elections which allow any registered voter to vote in a primary.
The current question being asked: is Cruz eligible to be on a particular state’s primary ballot MUST be answered by that state’s courts.
IF and ONLY IF the state court rules NO and Cruz claims the decision violates one of his US Constitutional rights could the appeal be heard in federal court and federal law applied.
For example if state X turns Cruz down for not being born on native soil, fine. But if it is claimed that it is because Cruz is of Canadian national origin, as opposed to just a foreign birth it would be cause for appeal to the federal courts. National origin is one of the constitutionally protected forms of discrimination.