WASHINGTON—Republicans rallied around embattled Supreme Court nominee Brett Kavanaugh, vowing Monday to push his confirmation through the Senate even as a new allegation of sexual misconduct emerged days before a hearing on an earlier assault claim.

Senate Majority Leader Mitch McConnell (R., Ky.) took to the Senate floor to call the allegations a “smear campaign” and promise a vote on the Senate floor to confirm the judge “in the near future.” President Trump, in New York for a United Nations meeting Monday, reiterated his...