Working under a suspension of the rules, which requires a two-thirds majority, the House rejected a veterans affairs funding bill 264-164, a national parks bill 252-176 and a District of Columbia funding bill 265-163, Washington publication Roll Call reported.
The Republicans managed to lure no more than 34 Democrats to their side on any of the three bills.
House Rules Committee Chairman Pete Sessions, R-Texas, said his panel would try again Wednesday, using a different approach so Republicans could pass the measures with a simple majority, Roll Call said.
Party leaders announced the piecemeal spending strategy to their members during a closed-door meeting at the Capitol Tuesday afternoon, about 13 hours after the federal government shuttered because of a budget stalemate between the GOP-led House and the Democrat-led Senate.
"President Obama and [Senate Majority Leader] Harry Reid would rather shut down the government than engage in talks to end special treatment for the well connected under Obamacare," a House GOP leadership aide said. "Closed parks and delayed veterans benefits can be easily remedied and allow us to remain focused on forcing the Senate to finally come to the table and talk with us."
The White House said Obama would veto the piecemeal approach if the bills reached his desk and his press secretary, Jay Carney, said the GOP approach "shows the utter lack of seriousness that we're seeing from Republicans."
"If they want to open the government, they should open the government, and then we can negotiate about how we fund our budget priorities in the future," he said at the daily briefing with reporters. "A piecemeal approach to funding the government is not a serious approach any more than it would be a serious way to try to deal with the consequences of default and the absolute necessity to maintain the full faith and credit of the United States."
"How does the White House justify signing the troop funding bill, but vetoing similar measures for veterans, National Parks, and the District of Columbia?" Roll Call reported Michael Steel, a spokesman for House Speaker John Boehner, R-Ohio, said in a statement. "The president can't continue to complain about the impact of the government shutdown on veterans, visitors at National parks and D.C. while vetoing bills to help them.
"The White House position is unsustainably hypocritical," Steel said.
During debate Tuesday afternoon on funding the National Park Service, Minority Leader Nancy Pelosi, D-Calif., said Republicans were engaging in "cherry picking intended to isolate" the Affordable Care Act.
"This isn't about the parks," she said. "They're using the parks as a pawn to defund the Affordable Care Act.
"It's not going to happen."
Pelosi said Republicans were "throwing us crumbs while they take the pie."
Rep. Mike Simpson, R-Idaho, responded: "You think we're holding the Affordable Care Act hostage? You're holding the national parks hostage."
Democrats said even if Congress enacts a bill reopening the national parks, 81 percent of the Interior Department's operations would remain shut down.
Reid called the piecemeal approach "another wacky idea" and said individual funding bills will go nowhere in the Senate.
Speaking on the Senate floor, Reid said the tactic shows House Republicans "understand finally the government is shut down. But now they are focusing on trying to cherry pick some of the few parts of government that they like."
"They don't like it all but they like a few parts of it. It's just another wacky idea from the Tea Party-driven Republicans," he said. "You can tell that the Tea Party still wants to keep the government shut down."
Earlier Tuesday, House Republicans took photos with their appointees to a conference committee they want the Senate to join as a way of resolving the impasse over a stopgap spending bill. The Senate rejected the House's offer Tuesday morning.
The piecemeal approach also didn't find much love among Senate Democratic leaders, Talking Points Memo said.
Senate Majority Whip Dick Durbin, D-Ill., dismissed the plan.
"We're going to wait and see, obviously, what they have to offer," Durbin told reporters Tuesday. "But that Senator Ted Cruz [R-Texas] is now going to pick his favorite federal agencies to reopen? Come on. Let's get serious about this. There are a lot of agencies of government that need to be open. I'd suggest opening all of them."
Before the government shut down at midnight, Cruz had urged House Republicans to pass funding bills for individual agencies if Senate Democrats refused to defund Obamacare. Late last week Cruz and other Senate Republicans engaged in a 21-hour talk-a-thon on defunding Obama's signature healthcare policy.