It'll be interesting to see how this goes. I'd guess that he won't have a case against chess.com because all they did was publish information that was correct. Magnus Carlsen could be in for a rough time though.
If you read the filing, Nieman claims that chess.com made false statements, as well as conspiring with Carlsen and Nakamura against him. This will be very interesting indeed, and I'm looking forward to seeing what proof of cheating they can demonstrate, other than "the win seems improbable."