WASHINGTON—Republicans forged ahead with a hearing Monday to consider sexual-assault allegations against Supreme Court nominee Brett Kavanaugh and set a deadline of Friday to determine whether his accuser appears, after days of political wrangling over how best to handle accusations that threaten his nomination.

White House Spokesman Raj Shah said the president won’t look at naming any replacement nominee unless there is a clear need. The administration, he said, was going “full steam ahead” to support Judge Kavanaugh, who...