House lawmakers on Friday approved a proposal to address potential security breaches on the Web site as Republicans seek to keep attention on problems associated with the rollout of the federal health-care law.

On a vote of 291 to 122, more than five dozen Democrats joined with all voting Republicans to approve a measure that would require the Department of Health and Human Services to notify affected users of any potential breach on a state or federal exchange within two business days.

Republicans’ aides openly boasted Friday about the 67 vulnerable Democrats willing to buck President Obama, who strongly opposed the measure. The White House said the proposal would create “unrealistic and costly paperwork requirements” that wouldn’t improve the safety or security of the site’s users.

The vote capped the first week of the new year for the Republican-controlled House, but comes as attention on Capitol Hill has started shifting to broader concerns with income inequality, especially regarding unemployed or low-wage workers, and away from concerns with the health-care law.

House GOP leaders this week distributed talking points to rank-and-file members advising them how to show compassion for the nation’s unemployed. Even House Speaker John A. Boehner (R-Ohio) sought to remind reporters that House Republicans are “concerned with those having a difficult time trying to find a job” as he raised doubts about Democratic proposals to extend unemployment insurance.

But Boehner also has vowed to continue passing legislation that seeks to hamper implementation of the Affordable Care Act. He described the measure approved Friday as part of GOP attempts to protect Americans “from the consequences of this disastrous law.”

Next week, Republicans will turn to another proposal that would require the Obama administration to provide detailed state-by-state reports on the number of people enrolling for health-care coverage under the law. The proposal is billed as another common sense attempt to address public concerns and measure the effectiveness of the law.

Democratic aides privately conceded this week that dozens of Democrats in tight reelection contests will continue voting with Republicans on health-care-related bills in hopes of blunting GOP attacks on the Affordable Care Act and to demonstrate a willingness to seek out bipartisan solutions to fix the law.

The bill approved Friday came as House Republicans have seized on the revelation that the chief information security officer at the Centers for Medicare and Medicaid Services, Teresa Fryer, recommended against authorizing the launch of on Oct. 1 on the grounds that not enough testing had been done in advance.

Fryer gave the House Oversight and Government Reform Committee a Sept. 24 memo she had drafted — but not sent — to her boss suggesting the federal health insurance marketplace “does not reasonably meet the CMS security requirements” intended to minimize risks. “There is also no confidence that the Personal Identifiable Information (PII) will be protected.”

Ultimately CMS administrator Marilyn Tavenner and the agency’s chief information officer Tony Trenkle, who has since retired, made the decision to launch as scheduled.

House Majority Leader Eric Cantor (R-Va.) read passages from the draft memo on the House floor Friday. But House Democrats argued that Republicans have exaggerated the Web site’s problems. They argue that some issues arise because consumers provide limited personal information when signing up through and because CMS put additional security measures in place a few days after Fryer voiced her objection to the launch.

“Republicans are still obsessed with killing this law,” Rep. Elijah E. Cummings (D-Md.) said before the vote Friday. “Since they cannot do so legislatively, they have shifted to a different tactic: scaring people away from the Web site.”

CMS also said Friday that there have been no successful security attacks on the Web site and that “no person or group has maliciously accessed personally identifiable information from the site.” The site undergoes security testing on an ongoing basis and is monitored to deter or prevent any authorized access, the agency said in a statement.

Even though the bill passed the House easily Friday, it will be ignored by the Democratic-controlled Senate, where leaders spent this week seeking to shift attention away from years of spending battles and the botched rollout of the health law to renewing unemployment benefits and raising the federal minimum wage.

“It’s a different political structure than it was even a year ago,” Sen. Charles E. Schumer (D-N.Y.) told reporters this week. “Issues like the deficit and Obamacare are important, but helping average people, getting the economy going and creating jobs is now number one, and [Republicans] block things like unemployment insurance and minimum wage at their peril.”