The Governmentâ€™s chief scientific and medical advisers painted a grim picture on Monday when they warned there could be 50,000 cases a day by mid-October if the current growth in the rate of infection is not halted.
Boris Johnson is set to announce a raft of new measures in response to the surge in infection, including ordering pubs, bars and restaurants in England to close by 10pm each night from Thursdays, and encouraging Britons to work from home "if possible".
Watch: All you need to know from Boris Johnson’s coronavirus statement to MPs
During a segment on GMB, Mr Morgan demanded that Dr Hilary explain how the government reached its 50,000 cases a day figure.
"What are you basing that on?", he asked. "If the answer is 'doubling every seven days'... that means the entire country, if we do nothing, will have ended by New Year's Day."
Dr Hilary replied that it did not mean that.
Mr Morgan shot back that it did, and in an extraordinary exchange, the pair went back and forth saying "it does" and "it doesn't".
The GMB presenter said: "If you say it doubles every seven days, that's where you get to by January 1st."
Dr Hilary replied: "But things change in an epidemic."
Morgan's co-host Susanna Reid then added: "If cases are going to double then there is going to be a point when everybody eventually has got the virus."
Dr Hilary responded "no!" before explaining that behaviour will change as cases rise, preventing every single person in the country from catching the virus.
"No, it completely ignores herd immunity, as a factor, it completely ignores that people start to shield, lock themselves away," he said.
Asked again how experts got to the 50,000 figure, Dr Hilary said it was reached based on the current trajectory.
He explained: "That's based on what's happened in the last few weeks.
"They're presenting the science, they're presenting the figures. They're giving us the information and the knowledge on which to base some rational decisions."
Morgan and Dr Hilary also clashed over closing times for pubs.
"My question is, I don't understand the logic of saying we're going to shut pubs an hour earlier", Piers said.
"They're all abiding by the safety rules doing it the right way and not much difference between 10 and 11.
"People will just go earlier, they'll go out earlier, they'll go out from 7pm instead of 8pm."