Democrat Joe Biden branded election rival Donald Trump a "national embarrassment" on Wednesday for failing to explicitly denounce white supremacist groups, as the president moved to quell the storm sparked by his remarks on the debate stage.

The bitter adversaries returned to the campaign trail one day after their off-the-rails showdown in Cleveland made headlines less for the substance than for its unrestrained chaos.

The toxic shout fest -- with Trump constantly interrupting and Biden launching personal attacks -- even prompted the overseeing Commission on Presidential Debates to announce it would be imposing new measures to help moderators "maintain order" at the upcoming two debates.

After his assertive performance Biden launched a whistlestop train tour Wednesday through battleground states Ohio and Pennsylvania where he renewed his criticism of Trump.

"The president of the United States conducted himself the way he did -- I think it was a national embarrassment," Biden said in Alliance, Ohio.

He fiercely attacked Trump's failure to make a clear and forceful denunciation of white supremacist groups or the far-right Proud Boys, instead giving a shout-out to the male-only militia group by saying they should "stand back and stand by," and that the real problem is "far-left" extremists.

"My message to the Proud Boys and every other white supremacist group is: cease and desist," Biden said. "That's not who we are. This is not who we are as Americans."

Trump, in an apparent attempt to tamp down outrage over his comments, called on the group to "stand down".

"I don't know who Proud Boys are but whoever they are they have to stand down," Trump told reporters at the White House. "Stand down, let law enforcement do their work," he said. "Whoever they are, stand down."

- Trump 'misspoke' -
Biden also knocked Trump for failing to speak directly to the US public about their plight during the Covid-19 crisis, which has killed more than 206,000 Americans.

"Does your president have any idea or understand what you're going through? Biden asked. "Or does he just ignore you, look down on you, lie to you?"

After the president failed to explicitly condemn white supremacists, Senator Tim Scott, the only black Republican in the US Senate, said Wednesday that he believes Trump "needs to correct" his comments.

"White supremacy should be denounced at every turn. I think the president misspoke, and he needs to correct it," Scott told reporters.

"If he doesn't correct it, I guess he didn't misspeak," Scott added.

Trump spent much of the debate time interrupting Biden, accusing him of being controlled by the "radical left" and trying to get under his skin with comments about his son Hunter's past business affairs.

The former vice president mostly kept his composure and went toe-to-toe with Trump for the full 90 minutes, attacking his handling of the pandemic, race relations and the economy.

Tuesday's showdown in Cleveland allowed Biden, 77, to rebuff Republican claims that he doesn't have the stamina for the White House job.

Trump earlier had insinuated that Biden is too old for the job and his mental faculties have been diminished by age.

An exasperated Biden variously described Trump as a "clown," a "liar" and "Putin's puppy" while telling him at one point to "shut up, man."

Snap polls on the debate performances of the presidential candidates leaned towards Biden, who withstood a barrage of barbs from the 74-year-old Trump while delivering some of his own.

A CBS sampling of 1,039 likely voters who watched the televised debate had Biden edging Trump by 48 percent to 41 percent -- a margin similar to that of national polls heading into the November 3 election.

- 'Meltdown' -
While Biden's camp was praising their candidate's performance Wednesday, Trump took to Twitter to lash out at Fox News host Chris Wallace, the moderator, claiming he had joined forces with Biden and ganged up on him.

Biden responded with a tweet depicting Trump as a crying baby.

The Democratic speaker of the House of Representatives, Nancy Pelosi, said Trump's debate behavior demonstrated why she had recommended that Biden not debate him at all.

"You saw a political nervous breakdown, a meltdown," Pelosi said on MSNBC. 

Trump "has never respected the dignity of his office and he demonstrated that last night." 

The Cleveland debate was scheduled to be the first of three before the election but the acrimonious debacle led to calls by some commentators for the two others to be cancelled.

In an interview with CNN, Biden's running mate Senator Kamala Harris suggested it was unlikely that the Democratic nominee would bow out of the next debates.

"Joe Biden's never going to refuse to talk to the American people," Harris said.

Trump was to attend a fundraiser Wednesday and address an outdoor campaign rally in Duluth, Minnesota, a Midwestern state narrowly won by Hillary Clinton in 2016.

The next debate is scheduled for October 15 in Miami.