The subpoenas issued last year originally cited a broad array of documents, including wiretap requests and other materials involving confidential sources that Holder argued he was prevented by law from supplying. Issa narrowed the request in negotiations with Holder in recent weeks to focus on documents pertaining to decision-making on withdrawing the February 2011 letter.
However, Holder refuses to turn over materials containing internal deliberations, and he asked Obama to assert executive privilege over such documents before Wednesday's committee meeting.
Boehner said Obama's assertion of executive privilege proved White House involvement and indicated a cover-up, which Carney later rejected.
"The issue here is about after-the-fact internal documents that have to do with the executive branch's ability to operate appropriately and independently in response to congressional investigations and media inquiries," Carney said.
He noted Holder had ended Fast and Furious when he learned about it, and ordered an inspector general to investigate it. The inspector general's report has yet to be completed.
"Every piece of documentation that relates to the operation itself, if (the Republican) interest is in the operation, how it came about, its origination, how it was approved, why such a flawed tactic was employed -- that has been provided to congressional investigators," Carney said.
The ranking member of the Senate Judiciary Committee, Sen. Chuck Grassley of Iowa, called Carney's statement "hogwash."
"Through my investigation, I know there are reams of documents related to 'the operation itself' that the Justice Department refused to turn over to Congress," Grassley wrote in a statement.
Cummings said Republicans were being extreme with their contempt measure.
"You have, once again, the far right of this party pushing and pushing," he added, leaving Obama with no option but to assert his executive privilege. " ... These were deliberative documents that all attorney generals have held close and not released. This attorney general released about 1,000 of those documents voluntarily, and Mr. Issa wanted more and more."
Grassley said Obama's assertion of executive privilege raises questions, but "we know of no presidential involvement" in Fast and Furious.
What he wants, Grassley said, is to get to the bottom of who was behind the sting.
"I've only been trying to find out who at the highest level of government ... gave approval for this, so we can get them fired," he said.
At Wednesday's hearing, Democrats noted that a similar program called Operation Wide Receiver was started during the Bush administration. They complained that Issa prevented the panel's investigation from fully examining any possible connection between Wide Receiver and Fast and Furious.
Republicans repeatedly invoked the name of the slain border agent in demanding that Holder turn over all documents sought by the panel.
"At the heart of the congressional investigation into Operation Fast and Furious are disastrous consequences: a murdered Border Patrol Agent, his grieving family seeking answers, countless deaths in Mexico, and the souring effect on our relationship with Mexico," Issa said in a statement. "Congress has not just a right but an obligation to do all that it can to uncover exactly what happened and ensure that it never occurs again."
Terry's family issued a statement Wednesday that called for all of the documents sought by the committee to be turned over.
"Our son lost his life protecting this nation, and it is very disappointing that we are now faced with an administration that seems more concerned with protecting themselves rather than revealing the truth behind Operation Fast and Furious," the statement said.
Asked about the Terry family's statement Thursday, Carney recited the steps Holder took to end Fast and Furious and to cooperate with the investigation of the operation and said the inspector general has "full access to all documents" sought by Republicans.
"That is separate from an attempt by Republican members of Congress to try to score political points," he said.
Grassley said any accusation of political motivation is "baseless," and he listed his attempts during the Bush administration to subpoena records or hold officials in contempt.
If the House finds Holder in contempt, it is unlikely he will be prosecuted for criminal contempt, according to Alan Morrison, associate dean at George Washington University Law School.
"It would look like terrible overreaching to go for criminal contempt," Morrison said. The charge carries a penalty of $1,000 and up to one year in prison.
Instead, Morrison said, it is more likely the House would pursue civil prosecution in federal court.
CNN senior legal analyst Jeffrey Toobin said that while Holder may be embarrassed, he won't ultimately be found in contempt.