He doesn’t have to prove jack shit if the judge is ordering them to turn over their internal documents as part of the case. The judge might look them over, decide there’s nothing relevant, and dismiss the case, but it won’t matter. The damage will already be done because sources will be less likely to trust them in the future.

You’re right, it does. He’s pretty unlikely to win in court, but lawyers are expensive. I expect they’ll reach a settlement just to avoid paying lawyers. Journalists aren’t known for having deep pockets to pay legal fees, unfortunately.