Typically judge's orders for medical records come into play in routine civil matters, such as injury cases. This is only because HIPAA created an extra layer of privacy, and with the order medical facilities have no objection. It also appears in divorce cases. As far as whether or not a judge would grant discovery of Martin's records is above my pay grade, but it's not inconceivable. It certainly seems relevant if Martin claims emotional and mental injuries directly resulting from harm by Incognito. But it doesn't seem as if Martin's claim is against Incognito, it's more broadly against the Dolphins Organization and the NFL. Incognito would be a third party, and I would think that might limit his rights to any of that information. In addition, mental health records are different than regular medical records and are legally protected. You'd have more hoops to jump through. But theoretically...if a judge signed an order, because he deemed it relevant, then the doctor or facility has no choice but to comply, or appeal. If any of that did occur, I would think that only limited information would be necessary, to establish a timeline (i.e. dates of visits prior to the incident, and diagnosis), and not the files themselves.
All this seems to be de-escalating to a degree, and certainly both Incognito and Martin have to be careful what legal claims they make, because lawyers have a right to discovery, and they can depose individuals. Martin got paid, and Incognito had some limited lost income, and will likely play for another team, so maybe they want this to go away. The NFL has deep pockets and everyone may be satisfied in the end.