WASHINGTON — President Trump, under fire for comments that even members of his own party called racist, amplified his attacks on four Democratic congresswomen of color on Monday, saying that they hated America and that one of the first two Muslim women elected to Congress sympathized with Al Qaeda.

In an extraordinary back and forth from opposite ends of Pennsylvania Avenue, Mr. Trump appeared to revel in the viciousness of his brawl with the four progressive women who have become the young faces of the Democratic Party. He goaded them into a response from Capitol Hill in which they denounced the president’s rhetoric and his policies, charging that he was pressing the agenda of white nationalists from the White House.

“He’s launching a blatantly racist attack on four duly elected members of the United States House of Representatives, all of whom are women of color,” said Representative Ilhan Omar, Democrat of Minnesota and the target of Mr. Trump’s most outrageous charges. “This is the agenda of white nationalists, whether it is happening in chat rooms, or it is happening on national TV, and now it’s reached the White House garden.”

The exchange was the latest episode in a presidency in which Mr. Trump has skittered from condemnations of black athletes kneeling during the national anthem to insults lobbed at developing countries to a defense of protesters at a white supremacist march. But now Mr. Trump is going after members of the majority party in the House, capable of fighting back.