Biden, invoking his late son Beau, raised reporting in The Atlantic that Trump called military members "losers" and "suckers," but Trump used it as a chance to bring up Biden's other son, Hunter.

"I don't know him," Trump said of Beau, before attacking Hunter's history with drug addiction.

"That is not true," Biden said. "My son, my son, like a lot of people you know at home, had a drug problem. He has overtaken it. He's fixed it. He's worked on it and I'm proud of him. I'm proud of my son. He wasn't given tens of millions of dollars. That is totally discredited."

Asked if he would be willing to denounce white supremacists, Trump said he would be willing but skirted around making the statement.

"Sure. I'm willing to do anything. I want to see peace," Trump said. "What do you want to call them? Give me a name, give me a name."

"Proud Boys, stand down and stand by, but I tell you what. Somebody has got to do something about antifa and the left because this is not a right-wing problem. This is a left-wing problem," Trump added.

"Antifa is an idea, not an organization," Biden said, quoting Trump's FBI director.

On the topic of race in America, Biden said he believes there is a systemic injustice in the U.S. on education, work and law enforcement, before offering praise to the majority of police officers.

"Look, the vast majority of police officers are good, decent, honorable men and women. They risk their lives every day to take care of us. But there are some bad apples and when they occur -- when they find them -- hey have to be sorted out. They have to be held accountable," Biden said.

"It's a little bit like how this guy and his friends look down on so many people. They look down their nose on people like Irish Catholics like me, who grew up in Scranton. They look down on people who don't have money. They look down on people who are of a different faith. They look down on people who are a different color. In fact, we're all Americans," he continued.

When confronted by Wallace about the rise in violence this summer in cities across the country -- run by both Democrats and Republicans -- Trump continued to blame the Democrats.

"I think it's a party issue," Trump said, even though Wallace brought up Tulsa, Oklahoma, and Fort Worth, Texas, both Republican-run cities.

Biden finished by urging Americans that the only way we can move forward is by coming together.

"The only way we're going to bring this country together is bring everybody together," Biden said.

"We can take this on and we can defeat racism," Biden added.

TRUMP'S CLAIM: "And now, we're weeks away from a vaccine."

FACT CHECK: Most prominent public health experts have said that a vaccine will not be widely available until mid 2021.

The pharmaceutical company Pfizer has said they may have data from its late-stage trial by October. Moderna has said it could have data from its late-stage trial by November or December. From there, the data will need to be analyzed by the FDA to determine whether the vaccine is both safe and effective. Other companies in late-stage trials are further behind. AstraZeneca's late stage trial is still on hold, and Johnson and Johnson only just started its late-stage trial this month.

Even if a vaccine is authorized by the end of the year, that does not mean it will be widely available. At first, only limited supplies will be available, and they will be prioritized for the most vulnerable.

"By the time you mobilize the distribution of the vaccinations and you get the majority or more of the population vaccinated and protected, that's likely not going to happen until the mid- or end of 2021," Dr. Anthony Fauci, the nation's top expert on infectious diseases, told MSNBC's Andrea Mitchell earlier this month.

In a recent Senate hearing, CDC Director Robert Redfield echoed those sentiments, "I think there will be vaccine that will initially be available some time between November and December, but very limited supply, and it will have to be prioritized," but, he said, "If you're asking me when is it going to be generally available to the American public so we can begin to take advantage of vaccine to get back to our regular life, I think we're probably looking at late second quarter, third quarter 2021."

-- ABC News' Sony Salzman

Despite Republican successes in statewide races in Ohio since 2016, the Buckeye State has become increasingly competitive as Trump and former vice president Joe Biden prepare to face off in Cleveland for their first debate on Tuesday.

In recent weeks Trump and Vice President Mike Pence have traveled repeatedly to Ohio, where they defeated Hillary Clinton and Sen. Tim Kaine, D-Va., her vice presidential candidate, in 2016, 51% to 43%.

Republicans also performed well statewide two years ago, winning the gubernatorial race and other down-ballot contests.

“Ohioans showed up for President Trump in 2016 and we expect them to do the same this November,” Trump campaign spokeswoman Samantha Zager told ABC News.

The president’s campaign has had a permanent presence on the ground in Ohio since 2016, and said they have made more than 10 million voter contacts in the state, where they and other GOP groups continue to canvass on the ground during the coronavirus pandemic.

The Biden campaign, meanwhile, has relied on virtual and digital organizing in the state during the pandemic, and has hosted hundreds of events and held 450,000 conversations with voters over the last month. Following Tuesday's debate, Biden will take a train tour of the state on Wednesday.

Biden has delivered a sharp economic message focused on Trump's record, centered on Toledo and Youngstown, home to several car plants and many blue-collar voters who abandoned Democrats and voted for Trump for years ago.

He's run at least eight television ads in recent weeks, including a response to Trump's call for a ban on Goodyear, the Ohio-based tire company than ran in Akron, where the company is based.

In an election year reshaped by the coronavirus pandemic, it’s also playing host to a number of legal battles over access to ballot drop boxes, which are increasingly used by voters as an alternative to in-person voting or relying on the United States Postal Service, amid concerns about service delays.

Read more about the battle for Ohio here.

-ABC News' Ben Siegel and Adia Robinson