The Justice Department has told House leaders that Attorney General Eric H. Holder Jr.’s decision to withhold certain documents about a flawed gun operation from Congress is not a crime and he will not be prosecuted for contempt of Congress.

Deputy Attorney General James M. Cole explained the decision, which was expected, in a letter to House Speaker John A. Boehner (R-Ohio). The letter was released publicly Friday, just over a week after President Obama invoked executive privilege to withhold the documents.

In his letter, Cole said the decision not to prosecute Holder conforms to long-standing Justice Department practice in both Democratic and Republican administrations.

In May 1984, Theodore B. Olson, then assistant attorney general, wrote that U.S. attorneys are not required to refer congressional contempt charges to a grand jury or prosecute an executive branch official “who carries out the President’s instruction to invoke the President’s claim of executive privilege before a committee.”

In July 2007 and February 2008, Attorney General Michael Mukasey cited the Olson analysis in letters to House Democratic leaders, informing them that Justice would decline to press charges against White House chief of staff Joshua Bolten and White House counsel Harriet E. Miers, who were held in contempt after failing to appear before the House Judiciary Committee.

“Consistent with this uniform position and practice, the Department has determined that the Attorney General’s response to this subpoena issued by the Committee on Oversight and Government Reform does not constitute a crime, and therefore the Department will not bring the congressional contempt citation before a grand jury or take any other action to prosecute the Attorney General,” Cole wrote.

On Thursday, Holder became the first attorney general to be held in contempt of Congress after he withheld internal deliberative documents that Republican lawmakers demanded as part of an investigation into the “Fast and Furious” gunrunning operation.

The Justice Department’s decision concerns only criminal action, so the House can still pursue civil litigation against Holder. On Thursday, the House voted to authorize civil action, which paves the way for a federal court challenge to Obama’s decision to invoke executive privilege.

Republicans dismissed Cole’s decision and said the U.S. attorney for the District — who technically has the authority to prosecute Holder — should be able to do so, noting that he is already leading an investigation into the possible leak of classified information by Obama administration officials to reporters.

“If he’s neutral enough to independently look for leaks from the administration and not be conflicted, then he has to be equally independent in weighing the statute” approved by the House, said House Oversight and Government Reform Committee Chairman Darrell Issa (R-Calif.), who is leading the “Fast and Furious” investigation.

Staff writer Ed O’Keefe contributed to this report.