Texas Democrat Al Green filed the resolution after the House voted to denounce Mr Trump’s attacks aimed at four US congresswomen as racist.
But the measure failed to win enough support, with his fellow Democrats voting overwhelmingly against.
Mr Trump said the “ridiculous” attempts to impeach him were now “over”.
“This should never be allowed to happen to another President of the United States again!” he tweeted.
Later, at a rally in North Carolina, Mr Trump continued his attacks on the four non-white Democrat congresswomen – Representatives Alexandria Ocasio-Cortez, Ilhan Omar, Ayanna Pressley and Rashida Tlaib.
He was cheered on by the crowd as he again accused the women of hating America. Many began chanting “Send her Back! Send her back!”- echoing the “Lock her Up” chant his supporters used against Hillary Clinton in the run-up to the 2016 presidential election.
Ms Omar responded on Twitter by quoting Maya Angelou’s poem Still I Rise: “You may shoot me with your words… But still, like air, I’ll rise.”
Why attempt to impeach Trump now?
Mr Trump’s attacks on the women began at the weekend, with a series of tweets telling the then unnamed politicians to “go back” to their countries. All four of the congresswomen are US citizens.
His tweets were widely condemned as racist, and the Democrat-controlled House passed a symbolic resolution denouncing Mr Trump’s “racist comments that have legitimised fear and hatred of New Americans and people of colour”.
Mr Trump has tweeted he doesn’t “have a Racist bone” in his body.
Mr Green appeared to be hoping increasing anger against the president would give him the support he needed.
In his resolution, he argued Mr Trump had “brought the high office of the President of the United States into contempt, ridicule, disgrace and disrepute”.
Why did it fail?
The Democratic leadership has so far refused to initiate impeachment proceedings, with Speaker of the House Nancy Pelosi repeatedly saying she does not want to act until an “ironclad case” has been built against the president.
“We have six committees that are working on following the facts in terms of any abuse of power, obstruction of justice and the rest that the president may have engaged in. That is the serious path that we are on,” Ms Pelosi told reporters on Wednesday.
But while the motion was defeated – with 332 voting against – it did indicate growing support among Democrats for impeachment proceedings.
A total of 95 Democrats voted in favour of the measure – an increase on the support Mr Green got for his two previous attempts, in 2017 and 2018.
What is impeachment anyway?
In this context, to “impeach” means to bring charges in Congress which will form the basis for a trial.
The US constitution states a president “shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes or misdemeanours”.
The process of impeachment has to be started by the House of Representatives and only needs a simple majority to pass. The trial will be held in the Senate.
But here, a two-thirds vote is necessary for the president’s removal – and this milestone has never been reached in America’s history.