Minority Leader Mitch McConnell, R-Ky., said Obama has chosen to raise money in Hollywood rather than work with Congress.
The institutional split between a Republican-led House and Democratic-controlled Senate has added up to inaction, especially in a midterm election year with control of the Senate at stake.
House Speaker John Boehner, R-Ohio, underscored the political divide, describing any attempt by Senate Democrats to add parts of a year-old comprehensive immigration bill to the border security measure as a "nutso scheme."
Lawmakers have struggled to compromise on a handful of bills to deal with the nation's pressing problems, sometimes having to negotiate within their own ranks. It hasn't always been successful.
Lacking the votes on their $659 million border bill, Republican leaders postponed a vote and huddled with their rank and file to figure out a next step. Tea party unrest stirred by Texas Sen. Ted Cruz and outside groups had forced the leaders to schedule a companion vote on legislation that would block Obama from extending deportation relief to any more immigrants living here illegally.
It was evident that wasn't sufficient, leaving some Republicans nervous about returning home to their districts without a vote on a border bill.
"I think we've seen this game before," said Rep. Devin Nunes, R-Calif., who is close to House GOP leaders. "Any time the groups come out and start to score about these issues, then senators get involved and they start having meetings and then they all sit together and sing 'Kumbaya' and stop any progress."
White House officials have indicated plans to unilaterally expand the deportation-limiting program, perhaps to millions more people, because of the House's failure to act this year on a comprehensive immigration overhaul bill passed by the Senate. Republicans warn that that would provoke a constitutional crisis, and a few conservatives have said it would be grounds for impeachment.
Boehner told reporters it would "seal a legacy of lawlessness."
The White House issued a statement assailing House Republicans for their inaction on comprehensive legislation and then willingness to target a program that "has benefited more than 500,000 young people who are Americans in every way except on paper."
House Minority Leader Nancy Pelosi, D-Calif., complained of a "mess on the floor of the House" in its final days of work before the recess. She said Republican leaders added the companion bill "to sweeten the pie" for immigration reform opponents and "intensify the harm for children."
The fast-moving developments would seem to ensure House passage of the border bill that would allow migrant youths to be sent home more quickly and would dispatch National Guard troops to the border, yet do nothing to change the overall stalemate in Congress over the border crisis in South Texas.
The Senate's version of the bill a $3.5 billion package that also includes money for Western wildfires and Israel faces opposition from Republicans and some Democrats, who argue the measure amounts to a blank check for Obama with no policy changes.
That left no apparent path for a compromise bill to reach Obama's desk before Congress' recess, even as lawmakers in both parties said they wanted to act.
Voicing her opposition to the House bills, Rep. Nita Lowey of New York, the top Democrat on the panel, said she regretted "the consequences of our failure."
Congress is poised to send Obama legislation revamping the VA, with a Senate vote expected Thursday. Lawmakers also are working on a path forward for highway and transit projects. The sweeping, $16.3 billion VA bill would overhaul the scandal-plagued department after reports of patients dying while awaiting treatment and long delays in scheduling appointments.
The legislative effort came against the backdrop of a partisan House vote to sue Obama for unilateral changes in his signature health care law. Republicans accused him of shredding the Constitution, while Democrats described the vote as a veiled attempt at impeachment.
The near party-line vote on Wednesday was 225-201.
Determined to help Israel amid weeks of deadly fighting in Gaza with Hamas, the House and Senate also were expected to approve $225 million for Israel's Iron Dome missile defense system that intercepts short-range rockets and mortars.
Support for Israel is strongly bipartisan in Congress. Immigration, on the other hand, causes sharp splits.
With a day left before the government plans to start reducing federal highway aid payments to states, the House insisted on its version of a bill to keep the money flowing.
The House bill allocates $10.8 bill to keep highway and transit programs going through the end of May 2015.
The Transportation Department has said that by Friday the federal Highway Trust Fund will no longer have enough money to cover promised aid and states should expect an average reduction of 28 percent in aid payments unless Congress acts first.
Senate Democrats have said they will not allow that happen, and they are expected to accede to the House demands.
Associated Press writers Joan Lowy, Alan Fram, David Espo and Erica Werner contributed to this report.