House Oversight Committee Chairman Trey Gowdy said Tuesday he would not launch an investigation into sexual harassment and assault allegations against President Trump, saying it’s the job of the police and prosecutors to investigate crimes.

He said the alleged crimes involve state law and should be referred to the Justice Department, but acknowledged the Justice Department doesn’t have jurisdiction over violations of state law.

“Allegations brought for now or in the future that, while serious, do not allege violations of specific criminal statutes, should be referred to the Judiciary Committee, which has jurisdiction over allegations related to fitness for office and non-criminal matters,” Mr. Gowdy said in his letter addressed to the Democratic Women’s Working Group.

Dozens of Democratic lawmakers demanded earlier on Tuesday for Mr. Gowdy’s committee to investigate the claims of 17 women, who have publicly accused Mr. Trump of bad behavior such as forced kissing and groping.

“Sexual abuse will not be tolerated whether it’s by a Hollywood producer, a chef at a restaurant, a member of Congress or the president of the United States. No man or woman is above the law,” said Rep. Lois Frankel, a Florida Democrat who heads the working group.

Lawmakers said the investigation’s immediate goal is to get at the facts behind the accusations, but didn’t rule out eventual impeachment of Mr. Trump.

“History has shown us that we impeached a president for inappropriate sexual behavior in the past,” said Rep. Brenda Lawrence, Michigan Democrat.

In their letter, which the women said was also signed by a number of male lawmakers, the Democrats said Mr. Trump should have a chance to defend himself but said his own past statements appear to bolster the women’s accounts.

“The president has boasted in public and in crude terms that he feels at liberty to perpetrate such conduct against women. Subsequently, Mr. Trump apologized and called it ‘locker room talk.’ He has since called all his accusers liars,” they said in the letter.

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