In a lengthy statement, Engel accused Pompeo of an "alarming disregard for the laws and rules governing his own conduct and for the tools the Constitution provides to prevent government corruption."
The State Department rejected that claim, with a spokesperson telling ABC News the threatened resolution is "political theatrics" and "an unfortunate waste of taxpayer resources."
The charge is a fiery escalation in the fight between the top U.S. diplomat and the Democrat-led panel charged with oversight of his agency. The committee is investigating Pompeo for orchestrating the firing of the department's inspector general, for delivering a political speech to the Republican National Convention and for refusing to turn over documents provided to a Senate Republican-led probe of former Vice President Joe Biden.
The contempt resolution is based on Pompeo's refusal to provide the House committee with documents for that Senate probe of Biden and last fall's impeachment inquiry, Engel said: "He seems to think the office he holds, the Department he runs, the personnel he oversees, and the taxpayer dollars that pay for all of it are there for his personal and political benefits."
The Senate investigation started last November, focusing on Biden's son Hunter and his position on the board of Burisma, a Ukrainian energy company -- the same kind of probe President Donald Trump pressed Ukraine's president to launch in exchange for security assistance and a White House meeting.
Engel issued a subpoena in late July for the more than 16,000 pages of documents the department provided to the Republican-chaired Senate Finance and Senate Homeland Security and Governmental Affairs committees. In contrast, few were provided to the Democrat-chaired House committees during the impeachment inquiry.
In a letter to the committee dated Thursday, the State Department rejected Engel's subpoena and denied his accusation of partisanship in how it has turned over those same documents to the Senate.
The department's cooperation with the Republican-led probes is not "inappropriate or unlawful because you disagree with the merits of your colleagues' oversight priorities," acting Assistant Secretary for Legislative Affairs Ryan Kaldahl said in the letter.
In response to Engel's request for the same documents, Kaldahl also told him the department would only provide them to his committee "if you can confirm by letter that the Committee is, in fact, substantively investigating identical or very similar corruption issues involving Ukraine and corrupt influence related to U.S. foreign policy."
There has been no evidence of corruption by Biden or the Obama administration in its Ukraine policy. But the claim has persistently chased Biden's campaign, fueled by Trump's frequent attacks on Hunter Biden and by a pro-Russian Ukrainian politician, according to the Office of the Director of National Intelligence.
In an assessment released on Aug. 7, the Office of the Director of National Intelligence's National Counterintelligence and Security Center director said the Russian government is taking active "measures to primarily denigrate former Vice President Biden." Specifically, the statement said Andriy Derkach, a Ukrainian member of parliament, "is spreading claims about corruption -- including through publicizing leaked phone calls -- to undermine" Biden and the Democratic Party."
Derkach and his associates have said that he provided "evidence" to the staff of Sen. Ron Johnson, chair of one of the two Senate committees investigating the Biden's. Johnson has denied that's true.
Engel referenced the ODNI assessment in his statement Friday, accusing Pompeo of trying to drag the committee into "a smear of the President's political rival."
"I want no part of it. Under no circumstances will I amplify Putin's debunked conspiracy theories or lend them credence. And I won't stand by and see the Committee or the House treated with such disdain by anyone," Engel added.
Henry Kissinger was the last Secretary of State to face a contempt of Congress charge, also for failing to provide a committee with subpoenaed documents, but the charge was ultimately dropped.
More recently, the House Oversight Committee voted to hold Attorney General Bill Barr and Commerce Secretary Wilbur Ross in contempt, while contempt of Congress was one of the charges for which Trump was impeached.