Democratic Sen. Tom Harkin of Iowa called on House Republicans to "be responsible and forget about kid's games like picking up their marbles and going home or throwing a temper tantrum or shutting down the government because you can't get your way."
On Thursday, Boehner had to delay their plan to introduce a bill to raise the nation's debt limit after conservatives complained the proposed package failed to include enough budget cuts and significant changes to entitlement programs.
The Obama administration says the debt ceiling must be increased by October 17 to ensure the government can pay all its bills.
Boehner and his top lieutenants initially hoped to move ahead with their proposal to permit Washington to borrow more money to pay its bills as soon as Friday.
The initial proposal by House GOP leaders, which would raise the debt ceiling for a year, included a lengthy list of GOP priorities including a one year delay of Obamacare, provisions to roll back regulations on businesses, tax reforms, and approval of the Keystone XL oil pipeline.
However, conservatives wanted more.
"It definitely has a lot of goodies in it, things that arguably would grow the economy and would arguably would generate more revenue," GOP Rep. Mo Brooks of Alabama told reporters, adding he was undecided on whether to support it. "Washington has a spending problem and this debt ceiling bill does not address the problem."
Another Republican, Rep. Cynthia Lummis of Wyoming, told CNN that she was also undecided but wanted to see deeper budget cuts on the measure.
"I came here to cut spending and to reduce the size of the federal government, so when those opportunities arise I want to take advantage of them," Lummis said.
When asked Thursday about the scope of cuts, Boehner told reporters that "in this bill, we have spending cuts and we have issues that will help spur more economic growth. We think the balance is correct."
Obama said Friday that the GOP strategy amounts to threatening to "burn the house down simply because you haven't gotten 100% of your way."
"That's not how our democracy is supposed to work," he said, repeating his past insistence that he will not negotiate under threat of a U.S. default because Congress failed to increase the debt ceiling.
White House spokesman Jay Carney told reporters on Thursday a one-year delay in implementing Obamacare's individual mandate for people to obtain health insurance would undermine a key provision of the program that prohibits the denial of coverage due to pre-existing conditions.
"The fact is you have to make the system work," Carney said, adding people with pre-existing conditions won't be denied insurance under Obamacare "because of the expansion of the number of people who will be covered and participate in these marketplaces provided by the Affordable Care Act through the individual mandate."
Some House Republicans questioned the strategy of proceeding to the debt ceiling fight before Congress resolved the question on spending and the possible shutdown. They argued the GOP still had some leverage to force a change to Obamacare on that measure.
CNN Chief National Correspondent John King said Thursday that focusing on the debt ceiling was where House Republicans "wanted to wage this fight all along."
"They didn't want to get bogged down in the government shutdown fight, but a conservative revolt within the House Republican ranks forced them to get there," King said.
Analysts warn of severe economic impact from any doubt cast over whether the United States would fail to meet its debt obligations. A similar bout of congressional brinksmanship over the debt ceiling in 2011 led to the first-ever downgrade of the U.S. credit rating.