While the Justice Department has appointed a special counsel, the former F.B.I. director Robert S. Mueller III, to oversee the bureau’s Russia investigation, Mr. Mueller is not fully independent because he could be fired at the president’s direction.

One of the speakers who addressed the crowd, Terry O’Neill, the president of the National Organization for Women, said, “If, in fact, a hostile, foreign government has installed a puppet as president of the United States, that affects the entire mission of my organization to elect good, progressive, feminist candidates to office.”

Mr. Trump has dismissed suggestions that his election victory was aided by the Russians, calling such allegations “fake news.”

Some celebrities, such as the actors Debra Messing, Alyssa Milano and Javier Muñoz, gave money or other support to the march, and related events were held in other cities. Many of the protesters used the opportunity to try to draw attention to a longer list of grievances with the administration, holding signs about issues that had been the subject of previous demonstrations, like climate change, women’s rights, a proposed tax overhaul and science.