Speaking at a rally in Nevada on Wednesday, the Republican nominee said: “I don't care how sick you are. I don't care if you just came back from the doctor and he gave you the worst possible prognosis, meaning it's over. Doesn't matter. Hang out till November 8. Get out and vote.“
Mr Trump added: “I say kiddingly, but I mean it."
He also reassured dying people that after they are gone, “All we’re going to say is: ‘We love you and we will remember you always’”.
Mr Trump's lack of tact appears to have already damaged his chances of becoming US leader. In addition to having made controversial and offensive statements frequently throughout his campaign and having alienated swathes of voters as a result, he floundered in his first debate with his opponent Hillary Clinton on 26 September, reversing gradual gains he had been making on her in the polls.
Pundits say Ms Clinton clearly dominated the televised exchange and as a result Trump dropped significantly behind her in the most recent voter surveys.
On Wednesday a new Ispos-Reuters poll put Clinton six per cent ahead of Trump, one point higher than a CNN-ORC survey released on Tuesday. In several swing states she is believed to have overtaken him.
During the debate at Hofstra University in New York, Mr Trump continued to attack a former Miss Universe contestant, when Ms Clinton raised the barrage of offensive pre-dawn tweets he had sent the woman. The Democrat also highlighted Mr Trump's refusal to publish his tax returns.
Ms Clinton said: “You've got to ask yourself, why won't he release his tax returns? And I think there may be a couple of reasons. First, maybe he's not as rich as he says he is.
“Second, maybe he's not as charitable as he claims to be. Third, we don't know all of his business dealings but we have been told, through investigative reporting, that he owes about $650 million to Wall Street and foreign banks.
“Or maybe he doesn't want the American people, all of you watching tonight, to know that he's paid nothing in federal taxes because the only years that anybody has ever seen were a couple of years when had he to turn them over to state authorities when he was trying to get a casino license and they showed he didn't pay any federal income tax…”
At which point, Mr Trump interjected: “That makes me smart”.
He added: “I will release my tax returns, against my lawyer's wishes, when she releases her 33,000 emails that have been deleted.”
The Secretary of State also accused Mr Trump of racism, after he boasted that he forced the former president Barack Obama to produce his birth certificate to prove he was an American citizen.
Donald Trump's most controversial quotes
Donald Trump's most controversial quotes
1/14 On Isis:
"Some of the candidates, they went in and didn’t know the air conditioner didn’t work and sweated like dogs, and they didn’t know the room was too big because they didn’t have anybody there. How are they going to beat ISIS?"
2/14 On immigration:
"I will build a great wall — and nobody builds walls better than me, believe me —and I’ll build them very inexpensively. I will build a great, great wall on our southern border, and I will make Mexico pay for that wall. Mark my words."
3/14 On Free Trade:
"Free trade is terrible. Free trade can be wonderful if you have smart people. But we have stupid people."
PAUL J. RICHARDS | AFP | Getty Images
4/14 On Mexicans:
"When Mexico sends its people, they’re not sending their best. They’re sending people that have lots of problems. They’re bringing drugs. They’re bringing crime. They’re rapists."
5/14 On China:
"I just sold an apartment for $15 million to somebody from China. Am I supposed to dislike them?... I love China. The biggest bank in the world is from China. You know where their United States headquarters is located? In this building, in Trump Tower."
6/14 On work:
"If you're interested in 'balancing' work and pleasure, stop trying to balance them. Instead make your work more pleasurable."
7/14 On success:
"What separates the winners from the losers is how a person reacts to each new twist of fate."
8/14 On life:
"Everything in life is luck."
9/14 On ambition:
"You have to think anyway, so why not think big?"
10/14 On his opponents:
"Bush is totally in favour of Common Core. I don't see how he can possibly get the nomination. He's weak on immigration. He's in favour of Common Core. How the hell can you vote for this guy? You just can't do it."
11/14 On Obamacare:
"You have to be hit by a tractor, literally, a tractor, to use it, because the deductibles are so high. It's virtually useless. And remember the $5 billion web site?... I have so many web sites, I have them all over the place. I hire people, they do a web site. It costs me $3."
12/14 On Barack Obama:
"Obama is going to be out playing golf. He might be on one of my courses. I would invite him. I have the best courses in the world. I have one right next to the White House."
13/14 On himself:
"Love him or hate him, Trump is a man who is certain about what he wants and sets out to get it, no holds barred. Women find his power almost as much of a turn-on as his money."
14/14 On America:
"The American Dream is dead. But if I get elected president I will bring it back bigger and better and stronger than ever before and we will make America great again."
“Nobody was pressing it, nobody was caring much about it," Mr Trump said. "I was the one that got him to produce the birth certificate and I think I did a good job."
Ms Clinton replied: “[Mr Trump] has really started his political activity based on this racist lie that our first black president was not an American citizen.
“There was absolutely no evidence for it. But he persisted. He persisted year after year.”Reuse content