WASHINGTON — California Republican Kevin McCarthy secured a clear shot to becoming House majority leader on Thursday as his sole rival dropped his bid in a leadership fight that exposed deep fissures within the GOP.

Barring an unforeseen challenge, McCarthy is on a glide path to the No. 2 job in the House behind Speaker John Boehner, with elections slated for June 19. Earlier in the day, backers of the four-term congressman had spoken confidently about his prospects.

Texas Rep. Pete Sessions, in a statement late Thursday, said he had decided to abandon the race after it “became obvious to me that the measures necessary to run a successful campaign would have created unnecessary and painful division within our party.”

Sessions, who serves as chairman of the House Rules Committee, has no plans to seek the No. 3 job of whip, said his spokeswoman, Torrie Miller. Three others are seeking that post.

Within 48 hours of Rep. Eric Cantor’s lightning primary-election downfall, McCarthy and his deputies aggressively rounded up votes with a pitch to Southern Republicans and pointed private conversations on the House floor in a race that occasionally had the markings of a personality-driven contest for class president.