After Mr Biden said he would have “beat the hell out of” Mr Trump if they were both in high school together the President said Mr Biden would “go down fast and hard, crying all the way”.
Mr Trump said the vice-president to Barack Obama was trying to “act like a tough guy” but argued he was in fact both mentally and physically “weak”.
Mr Trump said the Democrat did not know him and warned him not to “threaten” him.
“Crazy Joe Biden is trying to act like a tough guy. Actually, he is weak, both mentally and physically, and yet he threatens me, for the second time, with physical assault,” Mr Trump tweeted.
“He doesn’t know me, but he would go down fast and hard, crying all the way. Don’t threaten people Joe!”
Crazy Joe Biden is trying to act like a tough guy. Actually, he is weak, both mentally and physically, and yet he threatens me, for the second time, with physical assault. He doesn’t know me, but he would go down fast and hard, crying all the way. Don’t threaten people Joe!— Donald J. Trump (@realDonaldTrump) March 22, 2018
The row started on Tuesday when Mr Biden criticised the President over comments he has made about women.
“A guy who ended up becoming our national leader said, ‘I can grab a woman anywhere and she likes it,’” Mr Biden said at an anti-sexual assault rally at the University of Miami.
“They asked me if I’d like to debate this gentleman, and I said no. I said if we were in high school, I’d take him behind the gym and beat the hell out of him.”
It is not certain what Mr Trump was alluding to when he said Mr Biden was “weak” but it could be a reference to times when the former Delaware senator has become emotional during public appearances.
Mr Biden burst into tears after Mr Obama awarded him with the highest civilian honour in the US during a surprise ceremony in January 2017.
Mr Biden turned his back to the crowd to wipe away tears after the former president announced he was bestowing him with the Presidential Medal of Freedom.
The former vice-president has also spoken emotionally about his son’s brain cancer diagnosis and subsequent death. He previously said his son’s illness made him feel unable to run for president against Mr Trump.
“No one should ever seek the presidency unless they are able to devote their whole heart and soul and passion into just doing that,” Biden told ABC News in 2016. “Beau was my soul. I just wasn’t ready to be able to do that. My one regret is my Beau’s not here. I don’t have any other regrets.”
The death of Beau, an Iraq War veteran who died in 2015, is not the only tragedy to have hit the family. Beau was just two when his mother, Neilia Hunter, was killed in a car crash in 1972, which also took the life of Mr Biden’s one-year-old daughter.
Mr Biden, the first Roman Catholic vice-president, ruled out running for office in 2016 in a race that would have seen him pitted against Hillary Clinton and Bernie Sanders. He has since said he would have been able to beat Mr Trump if he had been the Democratic candidate. Mr Biden previously pursued the Democratic presidential nomination in 1988 and in 2008 but dropped out of the race early on both occasions.
Mr Trump’s stern warning to Mr Biden on Thursday is reminiscent of the time he implied he would be willing to fight Mr Biden behind a barn at a rally in Florida in 2016.
Referring to a fist fight between the pair, Mr Trump said: “I’d love that.”
He branded Mr Biden “Mr Tough Guy” and said of beating him up: “Some things in life you could really love doing.”
The comments followed Mr Biden saying he wished he was in high school with Mr Trump and could “take him behind the gym”. He said: “The press always asks me, don’t I wish I was debating him? No, I wish we were in high school and I could take him behind the gym. That’s what I wish.”
Mr Biden’s comments referred to the leaked tape from 2005 which surfaced in October 2016 in which Mr Trump could be heard bragging about groping and making unwanted advances on women. He boasted that he was such a “star” he can grab women “by the p***y”.
Mr Biden sought to clarify his remarks at a rally just two days afterwards, saying that he would only fight Mr Trump if he were still in high school. He added: “I want to make it clear I understand what assault is. I’m not in high school.”