The California Republican will have three weeks to try to tame the conservative opposition threatening to block his Oct. 29 election on the House floor. The timing is tough. Congress this month could consider a rush of contentious legislation—including a possible two-year budget deal and a debt-ceiling increase—likely to spark some GOP opposition.

Conservatives have made clear they will be weighing how Mr. McCarthy acts as they decide whether to stage an insurrection on the House floor. On Wednesday, the House Freedom Caucus, a group of about 40 conservatives, said at least 30 of its members would vote for Rep. Daniel Webster (R., Fla.) as speaker on Thursday and later on the House floor, unless Mr. McCarthy pledges to overhaul how the chamber is run.

“Whoever wins tomorrow has three weeks to make those changes,” Rep. Raul Labrador (R., Idaho) said Wednesday. This week’s vote, he said, “won’t settle anything.”