Why would you think that? None of these ruling have had anything to do with the merits of the case. They dealt with "standing" and with requests for "stays and injunctions", to prevent actions from occurring, thus preventing harm until the Court could rule. Many of the cases are still alive, even if their requests for injunctive relief and/or stays have been rejected.
Even at the lower court levels, the cases have not been argued, other than at the level of "briefs". Evidence has not been presented, if there is any to present.
I think that because if there was any question about his eligibilty, then I think at least one of the justices would step up and let their concern be known.