Kelly Shackelford, the president of the First Liberty Institute, said the group is "grateful" for Cruz's opposition.

"Elevating Judge Ozerden to a lifetime appointment on an influential federal appeals court would be a mistake. ... There are many highly qualified candidates for the Fifth Circuit. Judge Ozerden is not one of them," Shackelford said.

Graham, who has said he supports Ozerden, hasn't yet scheduled a vote on the nomination. A spokesperson for Graham said on Thursday that they didn't have any announcements about a potential vote.

A spokesperson for the White House didn't immediately respond to a request for comment.

Republicans have put a premium on confirming Trump's judicial nominees, which they view as a unifier for the party. They've also set a record for their pace of confirming influential appeals.

But Cruz isn't the first GOP senator to come out against one of Trump's picks.