Meanwhile, the Senate Intelligence Committee has subpoenaed Donald Trump Jr, one of Mr Trump’s sons, to legally force him to testify. It is the first known legal summons issued to a member of the president’s family in connection with the investigation.
Mr Trump Jr will be expected to answer questions about testimony he gave to the Senate Judiciary Committee in September 2017, which was later contradicted by the president’s former lawyer Michael Cohen, US media report.
He will also probably be quizzed on his connections in Russia.
Why was there a contempt vote on Barr?
Democratic lawmakers put forward the measure after Mr Barr did not comply with a legal order to release the Mueller report without the redactions.
The 448-page report was released last month with parts blacked out, including information that is classified or linked to pending investigations.
After the legal order, Mr Barr formally asked the president to assert his right to executive privilege to stop the unredacted version of the report being released.
The judiciary committee voted 24-16 along party lines to refer a contempt citation against the attorney general for a full House vote. It was not clear when this vote would happen.
Democrats say they need to see the full report and its underlying evidence to investigate possible obstruction of justice by Mr Trump.
House judiciary chairman Jerrold Nadler said the vote was a “grave and momentous step”, calling the Trump administration’s refusal to provide the full report to Congress “an attack on the essence of our democracy”.
“We have talked for a long time about approaching a constitutional crisis, we are now in it,” the New York Democrat told reporters.
Earlier, Doug Collins, a leading Republican on the committee, said Democrats were acting out of anger and fear “without any valid legislative reason”.
A conflict reaching new levels of acrimony
This is hardly the first battle between a president and his political opponents in Congress. Barack Obama, George W Bush and Bill Clinton all responded to some congressional subpoenas with claims of executive privilege. The Republican-controlled House once even held Mr Obama’s attorney general in contempt, setting a perhaps soon-to-be-revisited precedent.
The stakes in this battle, however, are as high as they’ve ever been, and the tone of the conflict has reached new levels of acrimony.
At issue isn’t just an unredacted version of the Mueller report, which could contain damning, previously undisclosed details or amount to a big nothing. It’s the entire sweep of congressional oversight of the Trump administration, including review of administration actions and Mr Trump’s business empire and financial ties.
The president has promised to resist all subpoenas, and Congress is contemplating legal action, further rebukes or even impeachment. Some are suggesting assessing hefty fines on intransigent administration officials or invoking the seldom-used concept of “implicit contempt”, which in theory could culminate in the House sergeant-at-arms trying to arrest and imprison the attorney general.
It seems far-fetched, but such is the current state of executive-legislative relations. And neither side appears interested in backing down.
Could the attorney general be charged?
For Mr Barr to actually face the prospect of charges, the entire House – where Democrats hold a 235-197 majority over Republicans – would have to vote against him.
The measure may be largely symbolic as few expect the Department of Justice to indict its own head with criminal contempt.
However, it would send a powerful message to the White House that congressional Democrats will not retreat in an increasingly vitriolic showdown.
Mr Barr is the first attorney general held in contempt of Congress since the Obama administration’s Eric Holder in 2012.
How did the White House respond?
Earlier, the White House accused Democrats of a “blatant abuse of power” and blocked access to the full Mueller report by claiming executive privilege.
It allows a president to keep certain materials private if disclosing them would disrupt the confidentiality of the Oval Office decision-making process.
Press Secretary Sarah Sanders said the president had taken such a step at the attorney general’s request to reject “unlawful and reckless demands” by Democrats.
“They didn’t like the results of the [Mueller] report, and now they want a redo,” she said in a statement.
The judiciary committee chairman argued the White House was misapplying the doctrine of executive privilege.
Mr Nadler said: “This decision represents a clear escalation in the Trump administration’s blanket defiance of Congress’s constitutionally mandated duties.”
Mr Barr was greeted with a standing ovation as he entered a cabinet meeting on Wednesday.
Will Robert Mueller testify?
The House judiciary committee has formally requested Mr Mueller testify, though a date has not been set.
The special counsel is still a Department of Justice employee, which means the attorney general could prevent him from testifying.
Mr Barr has previously said he would not mind Mr Mueller testifying before Congress, but Mr Trump has said the special counsel should not appear before lawmakers.
When Mr Mueller handed in his report, a spokesman said he would be leaving the department “within the coming days”.
If he does quit, he will become a private citizen and able to testify regardless of the department’s wishes.
The panel’s investigation into alleged Russian meddling in US politics has been rumbling on for the past two years.