When confronted by Wallace about the disparity between what his public health officials are saying and what he has said about the timing of the vaccine, Trump stuck with his previous comments.

"We can have it a lot sooner," Trump said. "It's a very political thing because people like this would rather make it political than save lives."

Top U.S. government public health officials and other experts have said that it's possible -- if everything goes well, but unlikely -- that vaccine trials will have enough clear and compelling data to be available by the end of October or beginning of November. Dr. Anthony Fauci, the nation's foremost infectious disease expert, has said he thinks there will be a "safe and effective vaccine" by "the end of this calendar year."

Fauci has also said that the United States will not return to "normal" until the middle of next year.

Biden interrupted the president by referring to Trump's previous statements earlier this year, when he said the virus would be gone by Easter and then also referenced Trump's potentially dangerous comment that people can inject bleach to get rid of the virus. Trump then responded that he said the bleach comment "sarcastically" made.

Biden turned to the hundreds of thousands of Americans who have died due to the coronavirus.

"His own head of the CDC said if we just wore masks between now -- if everybody wore masks and social distanced -- between now and January, we probably save up to 100,000 lives," Biden said.

Wallace pivoted the conversation to the global COVID-pandemic, and Biden began by listing statistics including 200,000 Americans dead and over 7 million infected.

"The president has no plan. He hasn't laid out anything. He knew all the way back in February how serious this crisis was. He knew it was a deadly disease," Biden said. "And I laid out again in July what we should be doing."

You should get out of your bunker and get out of the sand trap and your golf course and go in your Oval Office and bring together the Democrats and Republicans and fund what needs to be done now to save lives," he continued.

Trump countered by saying if the country had listened to Biden then millions of people would have died, not 200,000, and touted his decision to restrict travel from China in February, along with his efforts on ventilators and vaccine development.

"When you talk about numbers you don't know how many people died in China, you don't know how many people died in Russia, you don't know how many people died in India. They don't exactly give you a straight count," Trump said.

While on the topic of the Supreme Court and the fate of the Affordable Care Act, Trump was quick to call Biden's health care plan "socialized Biden."

Biden, notably, has said he does not support "Medicare for All" and responded by calling out Trump for promising to repeal and replace Obamacare -- which he hasn't.

"He does not have a plan," Biden said. "The fact is this man doesn't know what he's talking about."

Less than 20 minutes into the debate, both candidates have interrupted each other multiple times, making it difficult to understand what either is saying. Wallace has specifically asked Trump multiple times to let Biden finish his thought.

Eventually, the cross-talks continued for long enough that the former vice president said, "Will you shut up, man?" to the president.

TRUMP'S CLAIM: Can you imagine where they say you have to have your ballot in by Nov. 10? Nov. 10. That means that's seven days after the election, in theory, should have been announced. We have major states with that -- all run by Democrats, all run by Democrats. It's a fraud. It's a rigged election.

FACT CHECK: It is true that mail ballots arriving after Nov. 3 will be counted in several states, but these ballots are not fraudulent.

For these mail ballots to be counted, they must be postmarked on or before Election Day, and in some states, they must be postmarked by the day before the election.

In a normal election year, 19 states accept ballots that arrive after the election, as long as they are properly postmarked and arrive within a state-specific timeline, between three and 14 days post-election, according to the National Conference of State Legislatures. These states are not "all run by Democrats" either. Alaska, Iowa, Maryland, Mississippi and Ohio -- all states with Republican governors currently -- have postmark-by deadlines on the books, even when there isn't a pandemic causing widespread changes to voting.

More states will have postmark-by deadlines for the November election, due to both court rulings and changes implemented by state officials.

In Wisconsin on Tuesday, for example, a federal appeals court upheld a lower court's ruling, which ordered county election officials to accept ballots received up to six days post-election as long as those ballots were postmarked on or before Nov. 3. Other battleground states where court rulings have extended the deadline for mail ballots to arrive include Michigan, Pennsylvania and Georgia, but the extension is being challenged in all three states.

In Kentucky, Democratic Gov. Andy Beshear and Republican Secretary of State Michael Adams reached a bipartisan agreement in mid-August to accept ballots returned by mail through Nov. 6, as long as those ballots are postmarked on or before Nov. 3.

-- ABC News' Quinn Scanlan