RENO, Nev. — Former Vice President Joseph R. Biden Jr. on Wednesday night delivered his most forceful response yet to President Trump’s attacks on him and his son, accusing the Trump team of waging a campaign of “lies, smears, distortions and name calling” geared at knocking him out of the presidential race.
Mr. Biden made his comments during a campaign swing through Nevada after days of internal debate among his advisers over how best to refute unsubstantiated claims by the president and his personal lawyer that Mr. Biden improperly assisted his son’s business ventures in Ukraine and China.
The counterattack, people close to Mr. Biden said, was intended to demonstrate what he has been promising supporters for the past week: that he would not let Mr. Trump “hijack” his campaign by allowing the president’s narrative to take root. It was also intended to prove, in a walk-and-chew-gum kind of way, that he could multitask even when facing withering political fire — unveiling a serious policy proposal on guns early Wednesday in Las Vegas, then pivoting to a sharp political attack by nightfall in the north.
“Let me make something clear to Trump and his hatchet men and the special interests funding his attacks against me: I’m not going anywhere,” Mr. Biden told a crowd of about 500 at the Truckee Meadows Community College here.
“You’re not going to destroy me,” he said to cheers from supporters, a handful wearing “Impeach 45” jerseys. “And you’re not going to destroy my family. I don’t care how much money you spend or how dirty the attacks get.”
The American people, Mr. Biden said, “know me and they know him. The idea of Donald Trump attacking anyone’s credibility is a joke.”
Earlier in the day, speaking at a gun safety forum in Las Vegas, Mr. Biden blasted the president and jokingly pretended to mispronounce the last name of his personal lawyer, Rudolph W. Giuliani, who has communicated with Ukrainian officials on Mr. Trump’s behalf.
He said there was “zero, zero, zero” evidence that his family had done anything wrong.
Mr. Biden’s increasingly bitter fight with Mr. Trump comes at a potential tipping point in the 2020 Democratic presidential race. Until recently, Mr. Biden had been leading in most national polls since he entered the field last spring.
But Senator Elizabeth Warren of Massachusetts, fueled by a highly disciplined campaign, has surged in many early-voting states, and even Mr. Biden’s aides have conceded that he remains prone to missteps and gaffes.
A Monmouth University national poll released Wednesday showed Mr. Biden locked in a statistical dead heat with Ms. Warren, who garnered 28 percent support from Democratic and Democratic-leaning voters to his 25 percent, a difference within the survey’s margin of error.
At Wednesday night’s rally, Mr. Biden took a swipe at some of his Democratic opponents who have accused him of focusing too much on Mr. Trump and not enough on issues facing voters.
“A lot of my opponents say we have to do more than just beat Donald Trump,” he said. “I agree. We have to do more than beat Donald Trump. We have to beat him like a drum.”
Mr. Biden’s team has been alarmed by reports that Mr. Trump’s supporters plan to launch an aggressive advertising campaign to portray the former vice president, as the person who had acted improperly — despite the lack of any evidence to support that claim.
And Mr. Biden’s top advisers have been equally incensed by what they view as the news media’s willingness to air unsupported allegations by Mr. Trump and Mr. Giuliani.
The president and his lawyer have alleged that Mr. Biden, as vice president, pressured the Ukrainians to force out a top prosecutor in order to derail an inquiry into a Ukrainian company that paid his son, Hunter Biden. But Mr. Biden said Wednesday that he had been enforcing the demands of the Obama administration and other Western nations by seeking to replace the prosecutor with an official more committed to fighting corruption.
On Sunday, Anita Dunn and Kate Bedingfield, two senior Biden strategists, wrote to several major television networks asking them to stop booking Mr. Giuliani on their news programs, accusing him of spreading “debunked conspiracy theories.”
Mr. Biden amplified that argument on Wednesday, urging reporters to regard Mr. Trump’s statements, and his tweets, as not simply fodder for controversy but as a dangerous “abuse of power” that included enlisting foreign leaders as allies in his re-election effort.
Mr. Trump, he said, is “afraid of just how badly I would beat him next November.”
The president, in his own remarks on Wednesday, did not agree.
“I’d rather run against Biden than almost any of those candidates,” Mr. Trump told reporters at the White House.