WASHINGTON—The moment California college professor Christine Blasey Ford finished her testimony about an alleged sexual assault in the 1980s by Supreme Court nominee Brett Kavanaugh, Republicans realized his confirmation was at serious risk.

In the White House, aides gathered in Vice President Mike Pence’s office in the Senate Dirksen Building were “demoralized,” officials said. The same sense of dread spread in the Senate, as lawmakers texted each other about their concerns. “Honestly, I think had the hearing stopped right...