WASHINGTON — California Republican Kevin McCarthy secured a clear shot to becoming House majority leader 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 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, 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, 49, 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.

The votes next Thursday for majority leader and whip may well not be the end of it. Several Republicans asserted that next week’s action won’t quiet ambitious lawmakers or factions in the GOP caucus, and leadership contests after November’s national midterm elections could produce a brand new lineup.

Despite conservative discontent, Boehner’s job does not appear to be in serious jeopardy. But some lawmakers noted there was a limit to his security. “The speaker is speaker in 24-hour increments. Literally 50 guys can call a revolt,” said Rep. Tom Cole of Oklahoma, a Boehner ally.

Cantor suffered a stunning defeat to little-known college professor Dave Brat in Tuesday’s Virginia Republican primary, a race that underscored the rift within the GOP between pragmatic, establishment conservatives and farther-right contenders pressing for no-compromise ideological stances.

Another Texan with stronger bona fides in the conservative ranks, Rep. Jeb Hensarling, passed on the race earlier Thursday, saying: “After prayerful reflection, I have come to the conclusion that this is not the right office at the right time for me and my family.”

While the majority leader race narrowed to a single candidate, the contest to replace McCarthy as whip expanded with the addition of Rep. Marlin Stutzman of Indiana. Already seeking the post were Reps. Peter Roskam of Illinois and Rep. Steve Scalise of Louisiana.