A top House Republican announced an investigation Wednesday into how the White House allowed presidential aide Rob Porter to work with temporary security clearance while being accused of domestic violence, as White House Chief of Staff John F. Kelly came under increasing criticism for his role in the scandal.

Rep. Trey Gowdy, South Carolina Republican and chairman of the House Oversight and Government Reform Committee, sent a letter to Mr. Kelly demanding information on the White House security clearance process and asking whether the rules were followed in Mr. Porter’s case.

“The committee is investigating the policies and procedures by which interim security clearances are investigated and adjudicated within the executive branch, and the extent to which any security clearance issued to Porter comported with those policies and process,” Mr. Gowdy wrote.

The investigation was announced one day after FBI Director Christopher A. Wray contradicted White House explanations for its handling of Mr. Porter, the former White House staff secretary who has been accused of physically and verbally abusing his two ex-wives. Mr. Porter, who denies the accusations, resigned last week.

After days of silence, President Trump came out Wednesday against domestic violence.

“I am totally opposed to domestic violence of any kind. Everyone knows that, and it almost wouldn’t even have to be said,” Mr. Trump told reporters at the White House.

The president added with apparent impatience toward the media, “So now you hear it, but you all know.”

The White House has been saying that Mr. Porter was allowed to continue working because a background investigation for his security clearance was “ongoing.” But Mr. Wray testified to Congress on Tuesday that the FBI had completed its background investigation in July and finished a follow-up inquiry requested by the White House in November.

White House press secretary Sarah Huckabee Sanders said that although the FBI had completed its work, the West Wing was waiting for the White House personnel security office to finish its own review.

Lawmakers in both parties say the episode raises concerns that Mr. Porter, who routinely handled classified documents prepared for the president, could have been vulnerable to blackmail. Some critics also say the Porter case highlights a broader problem of as many as two dozen White House staffers working without security clearance, including Mr. Trump’s son-in-law and top adviser, Jared Kushner.

House Speaker Paul D. Ryan, Wisconsin Republican, told reporters, “If a person who commits domestic violence gets in the government, then there’s a breakdown in the system. That breakdown needs to be addressed.”

House Minority Whip Steny H. Hoyer, Maryland Democrat, said the oversight committee’s investigation needs to determine “whether White House personnel were improperly accessing classified materials while holding only interim security clearances.”

He said the administration has a serious problem in which “appointees of dubious qualification and background have been placed into positions of great national security sensitivity before the proper vetting process has been completed to ensure they can be trusted with sensitive information.”

Critics of the White House, and even some White House officials, have zeroed in on Mr. Kelly, who was responsible for handling Mr. Porter’s hiring and background check process. As late as last week — before a photo surfaced of one of his ex-wives with a black eye — Mr. Kelly was defending Mr. Porter as “a man of true integrity.”

Amid reports that Mr. Trump is sounding out confidantes about possible replacements for Mr. Kelly, one of those mentioned, House Majority Leader Kevin McCarthy, California Republican, said the White House had not approached him about the job.

“I have not spoken to the president about anything about a job, and I never have,” Mr. McCarthy said. “And there is no job opening.”

Asked by radio host Hugh Hewitt whether he would take the job if Mr. Trump asked him, Mr. McCarthy called Mr. Kelly “an amazing person” and said “we’ve got a lot of work still to do.”

“That wasn’t a ‘no,’” Mr. Hewitt said.

“But there is no job out there,” Mr. McCarthy insisted.

Vice President Mike Pence said he was looking forward to working with Mr. Kelly for “many months to come.”

The vice president acknowledged that the White House botched its handling of the Porter case.

“I think the White House could have handled this better, and I still feel that way,” Mr. Pence said. “That being said, any more counsel I have on this I’ll share with the president of the United States.”

Asked if Mr. Kelly should be terminated over the mishandling of the situation, Mr. Pence emphasized Mr. Kelly’s military service as a four-star Marine general and the loss of his son, who was also a Marine.

“There are very few Americans, or American families, that have served this nation more honorably or sacrificed more for this country than the family of Gen. John Kelly,” Mr. Pence said.

As questions and criticism in the media have persisted for more than a week, it’s unclear whether the scandal has hurt Mr. Trump with voters. A Morning Consult/Politico poll released Wednesday found voters split evenly on the president’s job approval, 47 percent to 47 percent — the first time in nine months that Mr. Trump hasn’t been in negative territory in the survey.

The poll was conducted from Feb. 8 to Feb. 12, as the Porter scandal was being covered widely in the media. The time frame also includes a period in which millions of workers were starting to see the benefits of tax cuts in their paychecks.

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