WASHINGTON—Since House Speaker John Boehner announced his plan to retire at the end of October, he has seemed relieved to be exiting the unruly chamber he’s struggled to run for the last five years. But his path to a life of golf and leisure for now is blocked.

With the selection of his successor postponed and his party thrown into disarray by the last-minute withdrawal of his heir apparent, California Rep. Kevin McCarthy, Mr. Boehner said Thursday he would stay until his replacement is selected. That may not happen soon.

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