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

Wallace's first question to both candidates was on the Supreme Court. He asked why Trump believes he is right in confirming a justice before the election and why Biden believes he is right in saying the decision should wait until after Election Day.

"I will tell you very simply, we won the election. Elections have consequences," Trump said. "We have the Senate, we have the White House and we have a phenomenal nominee respected by all top top academic good in every way, good in every way."

"And by the way, the Democrats, they wouldn't even think about not doing it if they have -- the only difference is to try and do it faster. There's no way they would give it up. They had Merrick Garland, but the problem is they didn't have the election so they were stopped."

When the question was given to Biden, he said that the American people have a right to say who the Supreme Court nominee should be and that that nomination should wait until the November election concludes as Americans have already begun voting.

"It's just not appropriate to do this before this election," Biden said. "Roe v. Wade and the Affordable Care Act could both be struck down by a conservative Supreme Court."

Multiple Senate Democrats are refusing to meet with Trump's nominee, Amy Coney Barrett, citing the impending election, and have expressed mounting concern with her stance on health care as the Supreme Court is set to hear arguments on the fate of the Affordable Care Act one week after the election.

Senate Republicans, meanwhile, are moving ahead with the nomination despite claims they made in 2016 about not confirming Garland during an election year.