I completely agree with Ginger.This isn't just about Paula allegedly having a relationship with a contestant and what that will mean for the future of her employment with the production. (Though I think we can safely assume that fraternization with contestants is an issue that is covered in judges' contracts.)
This is also about all the ancillary issues that arise when a judge has a relationship with a contestant. Every time that contestant moves on to the next round, that means another competitor has not been given that opportunity. If Corey's upward motion can in ANY WAY be credited to Paula's interference, then failed competitors are going to be LINING UP to sue (and rightly so).
These failed competitors will very easily be able to argue that if it weren't for Paula's consistently favourable televised reviews of Corey's performances, he would never have got as far into the competition as he did, thereby unfairly depriving some other contestant of his or her chance to win the big prize. And even leaving aside Paula's televised reviews of his performances (after all, she is pretty much uniformly complimentary of everyone), we still have the additional issues of her cash being used to promote his chances in the competition, and her expert coaching being privately provided to him.
I don't see that there is any way on earth that ABC would have broadcast the interview if there is any chance the accusations are untrue, because to broadcast untrue statements is to say to Paula and Fox, "Here, come and take many millions of our dollars in settlement of our libelling you." If Corey's statements are untrue, he has clearly seriously libelled Paula -- he has directly caused her professional reputation serious damage -- as well as unjustifiably calling into question the fairness of the judging processes of a major international talent competition that makes millions of dollars for its owners. If Paula does not immediately sue Corey (and ABC) for libel, then we can assume his charges are true. And on that basis, I would expect to see disciplinary action against Paula, most likely in the form of her "resigning" from the show "so as not to tarnish the experience for the contestants" or some such, and lawsuits against the production from any number of failed contestants from that season.
Of course the official press release says Corey's statements are "lies". If they are indeed lies, Fox and Paula are preparing libel lawsuits even as I type. But I rather expect they are not.