WASHINGTON—In the end, it came down to one of their own.

It was Arizona Republican Sen. Jeff Flake who forced his reluctant colleagues to heed Democratic calls for an investigation into sexual-assault allegations against Supreme Court nominee Brett Kavanaugh.

Mr. Flake, a White House antagonist, said he would support Judge Kavanaugh’s nomination only if the Federal Bureau of Investigation reopened a background check for one week to look at the allegations brought forward by Christine Blasey Ford, a California college professor. Other undecided senators soon endorsed that call, and Mr. Trump on Friday afternoon ordered the FBI to conduct such an probe.

On Friday morning, Mr. Flake provided the first sign of the impact of the previous day’s dramatic testimony before the Senate Judiciary by Dr. Ford and Judge Kavanaugh on the allegations, which Judge Kavanaugh denies. Despite his uncertainty, Mr. Flake said, he would support the nominee.

“I left the hearing yesterday with as much doubt as certainty,” Mr. Flake said early Friday. “What I do know is that our system of justice affords a presumption of innocence to the accused, absent corroborating evidence.”