No standing...
Actually, in that he is running against him, he should have standing, because he could be directly harmed if he were to lose the election to someone not eligible. General citizens however, do not have standing. This is where McCain screwed up against Obama. McCain could have brought suit, because he would have standing, but because of his own need to have Congress declare him eligible to run, he did not.
You would think that world-class lawyer Trump claims he has working on the suits would have told him by now. Yet he keeps threatening.