The showdown between Congress and the Justice Department over Russia escalated Tuesday after the House Judiciary Committee voted in favor of a resolution demanding that Deputy Attorney General Rod Rosenstein hand over sensitive documents related to the investigation into suspected meddling in the 2016 U.S. elections.

Committee members approved the resolution on a 15-11 party-line vote. It is now up to Republican leaders to decide whether they will bring the measure for a vote before the full House.

If the resolution passes, Mr. Rosenstein will have seven days to respond to the document request, but there is no penalty if he does not comply. Rep. Jim Jordan, Ohio Republican, promoted the measure as a first step toward punishing Mr. Rosenstein if he does not turn over documents requested by the committee.

“We want the full weight of the United States House of Representatives behind this resolution saying ‘we are entitled … to get the information we need to do our oversight duty,” Mr. Jordan said.

Options include subpoenaing Mr. Rosenstein, holding him in contempt of Congress or going so far as to impeach him.

Although impeachment is rarely used, conservative lawmakers have discussed it more frequently in recent weeks. In April, the House Freedom Caucus drafted eight articles of impeachment against Mr. Rosenstein.

House Republican leaders have said the documents should be handed to Congress, but they have not taken the Judiciary Committee’s hardball approach.

Speaker Paul D. Ryan said he expects the document requests will be “honored very, very soon.”

“If we do not get that, then we’re going to keep every single option available to us,” he said.

The Justice Department on Monday rebuffed a deadline to turn over the remaining Russia documents to Rep. Devin Nunes, California Republican and chairman of the House Permanent Select Committee on Intelligence.

Republicans have asked for documents related to the origins of special counsel Robert Mueller’s Russia investigation, the FBI’s use of confidential informants to make contact with Trump campaign figures and the bureau’s use of the now-discredited Steele dossier to obtain a warrant to monitor Trump campaign aide Carter Page.

In total, Republicans are seeking more than 1 million documents.

The Justice Department contends it has provided everything it can without jeopardizing confidential investigations.

Conservative lawmakers said that is not enough. They point to unanswered subpoenas and the slow pace of document production as evidence that the Justice Department is trying hide information from Congress.

“We are sick and tired of the Department of Justice giving us the runaround,” Mr. Jordan said Tuesday.

Democrats on the Judiciary Committee accused Republicans of using the document fight to undermine Mr. Mueller’s investigation. They say Republicans are trying to impugn Mr. Rosenstein, who oversees the Mueller probe, thus giving President Trump an excuse to fire him and end the investigation.

“This is an attempt to interfere with an ongoing investigation,” said Rep. Jerrold Nadler, New York Democrat. “It is not right. It may be constitutional, but it is not right for Congress to seek information about an ongoing investigation, especially when the obvious intent is to interfere with that investigation.”

Rep. Sheila Jackson Lee, Texas Democrat, wondered why Republicans are pushing a resolution that lacks penalties if Mr. Rosenstein does not comply.

Rep. Bob Goodlate, Virginia Republican and committee chairman, said the resolution is meant to call attention to the Justice Department’s failure to comply.

Mr. Goodlatte also appeared to show some skepticism toward the measure, even though he voted for it. He said he was unsure if the resolution was “completely necessary.”

Mr. Rosenstein and FBI Director Christopher A. Wray will appear before the committee on Thursday afternoon.

On Wednesday, FBI agent Peter Strzok, whose anti-Trump texts got him removed from the Russia investigation, will testify in a closed hearing.

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