WASHINGTON—House Republicans on Tuesday dropped their effort to curb the independence of a nonpartisan ethics board after a fierce backlash to it eclipsed other news on the first day of the new session of Congress.

Meeting behind closed doors on Tuesday, House Republicans unanimously decided to scrap their effort to place the independent Office of Congressional Ethics under the oversight of the House Ethics Committee, a move that would leave lawmakers policing themselves. That move, announced late Monday night, drew swift...

WASHINGTON—House Republicans on Tuesday dropped their effort to curb the independence of a nonpartisan ethics board after a fierce backlash to it eclipsed other news on the first day of the new session of Congress.

Meeting behind closed doors on Tuesday, House Republicans unanimously decided to scrap their effort to place the independent Office of Congressional Ethics under the oversight of the House Ethics Committee, a move that would leave lawmakers policing themselves. That move, announced late Monday night, drew swift pushback from government-watchdog groups, Democrats and some Republicans, who heard from angry constituents about the proposal.

President-elect Donald Trump, in tweets Tuesday mornings, questioned the timing of the move over other congressional priorities.

“It’s like a circular firing squad—our first day here and we’re passing around the handgun,” lamented Rep. Rod Blum (R., Iowa).

As criticism mounted Tuesday, Republicans decided midday to abandon the measure for now, though lawmakers said they would try to advance changes to the ethics watchdog later this year. Lawmakers have raised concerns over the board, including objections that it makes complaints against them public.

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Rep. Bob Goodlatte (R., Va.) pushed the ethics change, which would put the nonpartisan Office of Government Ethics under the control of the House Ethics Committee, which is run by lawmakers.Photo: Pablo Martinez Monsivais/Associated Press

House Republicans meeting Monday night had approved, by a 119-74 vote, the amendment from House Judiciary Committee Chairman Bob Goodlatte (R., Va.) to a package of new House rules. Both House Speaker Paul Ryan (R., Wis.) and Majority Leader Kevin McCarthy (R., Calif.) had objected to the amendment, urging a bipartisan approach to changing the office.

It wasn’t clear that the rules package would have had the votes to pass, given that it was likely to garner no support from Democrats, and some Republicans objected to the ethics amendment.

In two tweets Tuesday morning, Mr. Trump seemed to express sympathy with the move on its merits, calling the watchdog office “unfair.” But he said, “With all that Congress has to work on, do they really have to make the weakening of the Independent Ethics Watchdog, as unfair as it…may be, their number one act and priority.” He added that he would prefer a focus on issues “of far greater importance!”

The Office of Congressional Ethics serves as the chamber’s independent ethics watchdog by reviewing allegations against House members and staff. It is governed by an eight-person board of private citizens who don’t work for the government.

Democrats created the Office of Congressional Ethics in March 2008, more than a year after they took control of Congress, saying they wanted to help make the ethics process more transparent. If the office believes it has found violations, its job is to recommend a formal investigation by the traditional House Ethics Committee.

House Minority Leader Nancy Pelosi (D., Calif.) said Republicans’ reversal was a political calculation intended to change the discussion.

“House Republicans showed their true colors last night, and reversing their plans to destroy the Office of Congressional Ethics will not obscure their clear contempt for ethics in the People’s House,” Mrs. Pelosi said in a statement Tuesday. “Once again, the American people have seen the toxic dysfunction of a Republican House that will do anything to further their special interest agenda, thwart transparency and undermine the public trust.”

In the closed-door House GOP conference Tuesday, Mr. Goodlatte proposed stripping the ethics change out of the rules package, lawmakers said. Mr. McCarthy proposed taking up any changes separately through the House Judiciary Committee with the aim of finalizing any changes before the August recess, lawmakers said.

“Congress needs to do more to build the confidence of the American people—that this is their House,” said Rep. Walter Jones (R., N.C.), who praised the decision.

But other Republicans were furious, saying that the House needed to address complaints about the Office of Congressional Ethics. They criticized the office for acting on anonymous complaints and then making information public, forcing lawmakers to spend hundreds of thousands of dollars to defend themselves against charges without being afforded due process.

“Jesus had the right to face his accusers—he asserted that before the high priests, and we’re saying that members of Congress shouldn’t be?” said Rep. Steve King (R., Iowa.). He said that the office “unjustly targets members of Congress, and they should not be subjected to anonymous complaints that are acted upon by a body that then leaks that information.”

Under Mr. Goodlatte’s measure, the Ethics Committee would have been able to direct the office to stop any of its investigations into possible lawmaker misconduct. The measure would reduce how much information could be made public about investigations. The office, which was to be renamed the Office of Congressional Complaint Review, wouldn’t be able to publicly release any of its findings unless authorized by the Ethics Committee, and it wouldn’t be allowed to employ a press spokesman.

Under the amendment, the office would only be able to start an investigation if it received a “joint written request” to initiate one, but it wasn’t immediately clear who could make that request.

The office also wouldn’t be able to consider anonymous allegations against lawmakers.

Government watchdog groups, for their part, said the change would have diminished the office’s effectiveness in holding lawmakers accountable.

Before the OCE was created, the House Ethics Committee was a “black box of inaction,” Lisa Gilbert, director of Public Citizen’s Congress Watch, said in a statement.

“Following an election in which disdain for corruption in Washington was a defining issue—for the Republican presidential candidate, no less—it is an outrage that House Republicans are planning to undermine one of the modestly effective moves in recent years to reduce corruption,” she said.

Some worried the change would have made it easier for lawmakers to skirt ethics rules.

“If the 115th Congress begins with rules amendments undermining OCE, it is setting itself up to be dogged by scandals and ethics issues for years,” Norman Eisen and Richard Painter, chairman and vice chairman respectively of Citizens for Responsibility and Ethics in Washington, a nonprofit legal watchdog group, said in a statement.

Write to Kristina Peterson at kristina.peterson@wsj.com and Siobhan Hughes at siobhan.hughes@wsj.com