"There are not the votes in the House to pass" a clean spending bill, Boehner said, insisting that Democrats must accept some kind of alteration to the healthcare law before Republicans will agree to a budget deal.
"We voted to keep the government open," he said. "Providing fairness to the American people, under Obamacare, is all we're asking for."
On Tuesday, much of the federal government shut down when the current continuing resolution (CR) funding the government expired. House Republicans passed several bills shortly before the shutdown commenced that would have kept the government afloat while delaying or canceling key components of the healthcare law, but Senate Democrats killed each of the proposals in turn.
Since Tuesday, Republicans have called on Democrats to come to the bargaining table and offer some kind of concession on Obamacare in exchange for a bill restarting the government. Democrats, for their part, have said they are willing to entertain changes to the law, but not under the threat of prolonging the government shutdown.
Democrats are also not buying Boehner's insistence that a clean spending bill does not have the necessary support to pass the House, pointing to a contingent of House Republicans who have signaled a willingness to break ranks and join Democrats in supporting a clean spending bill if it means ending the government shutdown.
"The speaker said there aren't the votes on the floor to re-open the government," Sen. Chuck Schumer, D-N.Y., said on ABC after Boehner's interview. "Let me issue him a friendly challenge: put it on the floor Monday or Tuesday. I would bet there are the votes to pass it. We have just about every Democrat, 21 Republicans have publicly said they would. There are many more Republicans who have said that they privately would. So, Speaker Boehner, just vote. Put it on the floor and let's see if you're right."
On CBS' "Face the Nation", Treasury Secretary Jack Lew similarly pressed Boehner to let the House vote one way or another on a clean spending bill.
"Why doesn't he put it on the floor and give it a chance?" Lew asked. "You know, I worked for a speaker for eight years. I worked for Speaker [Tip] O'Neil who believed deeply that the one thing the American people won't tolerate is obstructionism. He put things on the floor and sometimes he won and sometimes he lost. But that's the right thing to do. Let the Congress vote."
This week, Democrats in the House began circulating a "discharge petition" that, if signed by a majority of House members, could force a vote on a clean spending bill over the objections of GOP leaders. To do that, however, they would have to persuade 18 Republicans to sign onto a proposal that publicly countermands their own leaders - no small feat.
And on Sunday, Rep. Peter King, R-N.Y., a centrist Republican who has criticized his party's approach to the budget debate - and exactly the kind of lawmaker whose support Democrats would need to push their petition over the top - said he would not support the effort to circumvent Republican leaders.
"It's not going to go anywhere," King said of the discharge petition on "Fox News Sunday." "There is no way in the world you're going to get 25 Republicans to go on that, and having said that, I wouldn't go out because [Democrats are], as I said, not bargaining in good faith here right now."
The protracted fight over the government shutdown has become conjoined with the fight over the nation's debt ceiling, which must be raised before October 17 if America is to avoid potentially defaulting on its debt, according to the Treasury Department.
On "This Week," Boehner threw down yet another gauntlet, this time about the debt ceiling, saying Republicans will not support a bill lifting the cap on the government's borrowing authority without any attendant deficit reduction.
"I don't want the United States to default on its debt," he said, "but I'm not going to raise the debt limit without a serious conversation about dealing with problems that are driving the debt up. It would be irresponsible of me to do this."
President Obama has said he simply will not negotiate over the full faith and credit of the United States, calling on Congress to raise the debt ceiling before any further fiscal negotiations begin. And that refusal to negotiate, Boehner said, is what's really pushing the United States to the brink of default.
"My goal here is to have a serious conversation about those things that are driving the deficit and driving the debt up," he said, "The president's refusal to sit down and have a conversation about this is putting our nation at risk of default."