WASHINGTON — Representative Nancy Pelosi and other top Democrats have been leery of any mention of the I-word. Clamoring for the impeachment of President Trump, they fear, could backfire and alienate independent voters and unhappy Republicans they hope to attract in next year’s midterms. But demanding congressional hearings into accusations of inappropriate sexual behavior by the president is another thing altogether.

With the rapidly shifting cultural and political environment on sexual misconduct evident in the defeat of Roy Moore in Alabama and the quick resignations of multiple lawmakers, Democrats — led by women in the party — are intensifying their efforts to draw attention to Mr. Trump’s own history of accusations made against him by a number of women who say he groped or sexually harassed and abused them.

“If you look at the mood of the country and the ‘Me Too’ movement, the time is really right to get to the truth of the matter,” said Representative Lois Frankel of Florida, the chairwoman of the Democratic Women’s Working Group. The group has collected more than 150 signatures from Democratic lawmakers on a letter demanding that Republicans convene hearings on the issue.

They are highly unlikely to get satisfaction on that front. Representative Trey Gowdy, the South Carolina Republican who leads the Oversight and Government Reform Committee, quickly rejected the request for a hearing and said he would refer the matter to the Justice Department.

But Democrats intend to press the issue in an effort to force Mr. Trump and the White House to respond to allegations he has long dismissed. They hope to underscore a contrast with Democrats like Senator Al Franken of Minnesota and Representative John Conyers Jr. of Michigan — two of their own high-profile lawmakers who announced that they would vacate their seats after accusations of sexual misconduct.

Democrats are taking varied approaches to try to make their case. Representative Luis Gutiérrez, Democrat of Illinois, used a House Judiciary Committee hearing on Wednesday to recount some of the accusations against the president while pressing Rod J. Rosenstein, the deputy attorney general, on whether the episodes merited investigation by the federal authorities.

“I will invite you to submit the evidence and the department will review it if you believe there’s a federal crime,” Mr. Rosenstein responded. “That applies to any alleged violation by any person. That’s all I have to say about that.”

Also on Wednesday, Representative Joseph Crowley of New York, the chairman of the Democratic Caucus, joined Representative Jackie Speier, Democrat of California, in asking the independent Office of Government Ethics to look into what they described as credible claims by at least 19 women.

“We believe public officials must set the gold standard for professional behavior, particularly those who hold the high offices of representative, senator, and — especially — president of the United States,” they said in a letter.

Mr. Trump seemed to anticipate the renewed emphasis this week when he declared in a tweet that Democrats, unable to prove campaign collusion with Russia, would resurrect sexual allegations made by women “who I don’t know and/or have never met.”

But in denying any misconduct and attacking Senator Kirsten Gillibrand, Democrat of New York and one of a half-dozen senators who have urged Mr. Trump to resign, the president seemed to only inflame the Democrats and increase calls that he leave office.

The tough Democratic line is a departure from the careful approach that leading Democrats have taken on the subject of impeachment. The majority of House Democrats opposed an attempt last week to initiate impeachment proceedings. Ms. Pelosi and Representative Steny Hoyer of Maryland, the No. 2 Democrat, issued a joint statement pointing to the special counsel looking at the Trump administration and campaign as well as current congressional investigations.

“Those inquiries should be allowed to continue,” their statement said. “Now is not the time to consider articles of impeachment.”

But Ms. Pelosi has strongly endorsed the push for new hearings on the sexual misconduct complaints against the president. “I don’t think that a person who has been a sexual harasser should be president of the United States,” she told CNN.

From a political perspective, the push for hearings over the president’s sexual conduct can serve as a middle-ground alternative to a more polarizing impeachment campaign. It allows Democrats to take aim at a potential vulnerability of the president, who bragged about his groping on a recording, in an environment where little tolerance is being shown for those who are accused, let alone investigated. It can provide some solace to frustrated voters in the Democratic base who don’t think the leadership is being aggressive enough in trying to push Mr. Trump from office. It also forces congressional Republicans and administration officials into the difficult position of either coming up with an explanation for not taking up the allegations or defending the president on a volatile issue.

At this stage, party strategists don’t expect Democratic congressional candidates to make the president’s past sexual behavior a central theme of next year’s campaigns. They prefer that the focus be on the economy, the Republican tax bill, health care and other issues with a more direct effect on voters. But they say that keeping the accusations front and center helps Democrats create an overarching atmosphere that plays to their advantage and particularly turns female voters against Republicans.

The White House and other Republicans say that Democrats should just accept the fact that voters knew of the allegations against Mr. Trump during the campaign and elected him anyway. But that was before the current furor over sexual harassment, and Democrats say the president, like other politicians, needs to be held accountable.

Though Republicans seem unwilling to convene a hearing that would put the president under harsh scrutiny, Democrats have no intention of dropping the subject.

“We are not going to let up,” Ms. Frankel said. “This is so much bigger than us.”

There is one sure way for Democrats to get the hearings they want — win back the House or Senate next November, install Democratic committee leaders and call witnesses. Democrats believe that a concentrated and persistent emphasis on longstanding accusations of sexual misconduct by Mr. Trump will help them do just that.