House Democrats said Monday they plan to introduce a resolution condemning President Trump’s “xenophobic” comments against four freshman Democratic lawmakers, as more Republicans publicly criticized Mr. Trump for his remarks telling the Democrats to go back to their original countries.

“The House cannot allow the president’s characterization of immigrants to our country to stand,” House Speaker Nancy Pelosi told colleagues in a letter. “Our Republican colleagues must join us in condemning the president’s xenophobic tweets.”

She said a resolution will be introduced by Rep. Tom Malinowski of New Jersey, who was born in Poland, and Rep. Jamie Raskin, Maryland Democrat.

“The president went beyond his own low standards using disgraceful language about members of Congress,” she said. “Our caucus will continue to forcefully respond to these disgusting attacks.”

In light of the tweets, Rep. Al Green announced he’d be pushing to force a vote on impeachment later this month.

“I will again, this month, bring impeachment to a vote on the Floor of the U.S. House of Representatives for bigotry in policy, harmful to our society,” the Texas Democrat announced.

While more than 80 Democrats are in favor of stating an impeachment inquiry, Mrs. Pelosi has been dead set against moving forward on that front.

Rep. Chip Roy, Texas Republican, became the first of several GOP lawmakers to criticize President Trump’s comments calling on a group of female freshman Democratic House members to go back to the countries they came from.

Mr. Trump “was wrong to say any American citizen, whether in Congress or not, has any ‘home’ besides the U.S.,” Mr. Roy tweeted late Sunday night.

But he added, “I just as strongly believe non-citizens who abuse our immigration laws should be sent home immediately, & Reps who refuse to defend America should be sent home 11/2020,” a reference to Election Day.

The only two black Republican lawmakers in Congress also criticized Mr. Trump’s comments. Rep. Will Hurd of Texas called the president’s remarks “racist and xenophobic;” Sen. Tim Scott of South Carolina said Mr. Trump engaged in “unacceptable personal attacks and racially offensive language.”

Rep. Mike Turner, Ohio Republican, said the president’s remarks “were racist and he should apologize.”

Two Michigan Republicans, Reps. Fred Upton and Paul Mitchell, also disagreed publicly with Mr. Trump on Monday. Mr. Upton said on Twitter that he was “appalled” by the president’s comments.

“There’s no excuse,” Mr. Upton said. “Inflammatory rhetoric from both sides of the aisle that is used to divide us just isn’t right. It’s not helpful. We have too many challenges facing us that we ought to be working on together – immigration, the debt ceiling, the border crisis. The President’s tweets were flat out wrong and uncalled for, and I would encourage my colleagues from both parties to stop talking so much and start governing more.”

Democrats have accused Mr. Trump of racism for his tweets that were apparently aimed at Reps. Ilhan Omar of Minnesota, a native of Somalia, and Reps. Alexandria Ocasio-Cortez of New York, Rashida Tlaib of Michigan and Ayanna Pressley of Massachusetts. All are U.S. citizens.

The president redoubled his criticism of the Democratic lawmakers for spewing “racist hatred” and anti-Semitism, and he challenged other Democrats to rebuke them.

“If Democrats want to unite around the foul language & racist hatred spewed from the mouths and actions of these very unpopular & unrepresentative Congresswomen, it will be interesting to see how it plays out,” Mr. Trump tweeted Monday. “I can tell you that they have made Israel feel abandoned by the U.S. When will the Radical Left Congresswomen apologize to our Country, the people of Israel and even to the Office of the President, for the foul language they have used, and the terrible things they have said. So many people are angry at them & their horrible & disgusting actions!”

Sen. Lindsey Graham, South Carolina Republican, said he spoke with Mr. Trump after the president’s initial tweets and urged him to tone down his rhetoric.

“Mr. President, you’re going to win. Just knock it down a notch,” Mr. Graham said on “Fox & Friends.”

He said the president needs to “aim higher” when talking about the liberal House Democrats.

“Talk about their policies,” Mr. Graham said, as if speaking directly to Mr. Trump. “They’re American citizens that are duly elected that are running on an agenda that’s disgusting. They’re anti-Semitic. They’re Socialist. Make them the face of the Democratic Party. They are American citizens. They won an election. You’re right about their policies. Just aim higher. Don’t get personal. Don’t take the bait. They’re on the wrong side of the future.”

The Trump campaign’s communications director said on Twitter that the president “loves this country & doesn’t like it when elected officials constantly disparage it & spew anti-Semitic rhetoric.”

“All Dems have leapt to defend the ‘Blame America First’ crowd when they really should be defending America & rooting out anti-Semitism in their ranks,” Tim Murtaugh said.

In an appearance on “Fox & Friends” on Monday, Mr. Roy said the president is “right” to be confronting the liberal House women over their opposition to the Immigration and Customs Enforcement agency.

“My radical leftist colleagues in the House, they need to go home, they need to be sent home, and I’m glad the president is taking them on,” Mr. Roy said. “The president is right to take them on, and we should make sure the American people know that that is the face of the Democratic Party. That is who needs to be sent home in 2020.”

• Tom Howell Jr. and Gabriella Muñoz contributed to this report.

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