WASHINGTON — After three years of cajoling, finessing and occasionally strong-arming his fitful conservative majority, Speaker John A. Boehner waved the white flag on Wednesday, surrendering to demands from his right flank that he tie money to keep the government open after Sept. 30 to stripping President Obama’s health care law of any financing.

Mr. Boehner knows that the plan he unveiled cannot pass the Senate, and that it may prove unwise politically and economically. His leadership team pressed just last week for an alternative. But with conservative forces uniting against him, he ultimately saw no alternative but to capitulate — and few good options to stop a government shutdown in two weeks.

“Today was a step forward, and a win for the American people,” said Representative Tom Graves, the Georgia Republican whose “defund Obamacare” push had amassed 80 House supporters, a bloc large enough to dictate the outcome.

With much of the government set to run out of money at the end of the month — and run out of borrowing authority by mid-October — Mr. Boehner faced a choice: he could steer a middle ground and find a way out of his fiscal dead end with Republican and Democratic votes, or he could yield to a conservative movement to strip the Affordable Care Act of financing, unite his Republican majority around that war cry, and hope for the best.