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Neither House Speaker Mike Johnson nor soon-to-be Senate Majority Leader John Thune had the power to call a recess and force through a nomination. While they both indicated they would allow that to happen, that was never the question because it wasn't up to them.
Instead, it would have taken a full vote by both chambers to recess. In the House, that can happen with a simple majority. In the Senate, a recess vote can be filibustered, making 60 votes the threshold. What that means is that Democrats and the Republicans who opposed Gaetz could easily block any attempt at a recess appointment.
But what about the theory that Trump could force a recess with executive power? That would have also been precarious because the Constitutional language specifically applies to a disagreement between the chambers on when to recess. To trigger that, you would first need both chambers to vote to recess on different days. That wasn't going to happen. It's also worth noting that three of the five conservative justices on the Supreme Court have already made it clear they believe recess appointments are unconstitutional, which means any challenge would have likely succeeded given the makeup of the court.