WASHINGTON—Amid the bare-knuckles brawl over the Supreme Court, former Obama administration lawyer Neal Katyal broke with many Democrats to endorse nominee Neil Gorsuch, and Republicans in turn gave Mr. Katyal an unusually prominent role by having him introduce the judge at his confirmation hearing.

Now, Judge Gorsuch’s confirmation Friday puts him on course to take the high-court bench for the first time on April 17, and he will see a familiar face at the lawyer’s lectern: Mr. Katyal, who has a case before the justices that...