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“For the majority leader and the Democrats in the House and the White House to say, ‘Absolutely, categorically not, we will not consider what the Republicans in the House of Representatives are doing,’ is piling on. It’s piling on, and it’s not right,” said Sen. John McCain, Arizona Republican.

House Republican leaders had hoped they’d found popular changes that would be able to win widespread support among the GOP and also earn backing from some Democrats who want to distance themselves from Obamacare.

Key among those changes was the plan to make Mr. Obama, Mr. Biden and their Cabinet officials have to participate in the health exchanges, without government subsidies, the same as many other Americans.

“If Obamacare is good for members of Congress, then it’s good for the president,” said Rep. Darrell Issa, California Republican.

House Democratic leaders said the GOP was being “irresponsible” by not just raising the debt outright, with a deadline looming.

“It appears that once again our House Republican colleagues are prepared to put the economy at risk to advance their political agenda,” Rep. Xavier Becerra, chairman of the House Democratic Caucus told reporters Tuesday morning.

Rep. Joseph Crowley, New York Democrat, predicted that the House Republicans’ bill will never pass the Senate and may not even pass the House.

Jacqueline Klimas contributed to this article.