Sure. Each state has it's own specific procedure, and they are pretty varied. But the loser has rights.
I will say that the extent of mess depends on the loser, fighting. Maybe Hillary or sanders or whoever will concede that Cruz is qualified.
-- The State party can challenge the slate (and has, actually) but what good would it do for a state Pubbie party to challenge it's own slate? --
The GOP has certified Cruz eligible. That precludes suing to find him otherwise, through the primaries.
:: The GOP has certified Cruz eligible ::
And, therein, lies the rub.
DJT should STHU, citing this simple fact.