Rounding up votes for a Republican budget blueprint the House approved Thursday, House Majority Whip Kevin McCarthy told GOP lawmakers that he needed the support of all Republicans—or almost all of them.

"Not everyone can be running for Senate in Georgia," he joked in a closed-door meeting this week, according to lawmakers and aides. In the end, Mr. McCarthy lost the votes of all three Georgia Republicans competing in a GOP primary for the Senate, as well as those of nine other Republican congressmen.

That left the House GOP with a narrow, 219-205 vote in favor of the budget—and an illustration of which lawmakers felt that backing their party's budget plan would be a political risk in the runup to the November elections. Drafted by House Budget Committee Chairman Paul Ryan (R., Wis.), the 10-year budget blueprint serves as Republicans' chief statement of policy priorities, a wish list of GOP proposals to reduce federal spending and overhaul safety-net programs.

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