Mr Trump is refusing to acknowledge the election was won by his bitter Democrat rival Joe Biden and is pursuing a number of legal actions in a bid to cling onto power. But Mr Morgan, who knows Mr Trump from his time as a broadcaster in the US, called on the President to face the music.
The GMB presenter tweeted: “Dear Donald, It’s over. You lost. Suck it up.
“Don’t be the Biggest Loser. Be the guy who fought hard, came up short – and left with dignity.
“Your friend, Piers.”
Veteran Democrat Mr Biden cleared the threshold of 270 Electoral College votes needed to win the presidency on Saturday, four days after the November 3 election.
But Mr Trump, who has attacked the integrity of the US voting process without offering evidence, has yet to accept defeat and some of his allies have encouraged him to exhaust every resource for hanging onto power.
He has launched an array of lawsuits to press claims of election fraud for which he has produced no evidence.
And his campaign spokesman Tim Murtaugh said he was planning to hold rallies to build support for his election challenge.
Asked when Trump would concede or call Mr Biden to congratulate him on his victory, Trump campaign adviser Jason Miller said: “That word’s not even in our vocabulary now.
“We’re going to pursue all these legal means, all the recount methods.”
Rudy Giuliani, Mr Trump’s personal lawyer and a former New York City mayor, tweeted: “The Biden selection by the Crooked Media is based on unlawful votes in PA, Mich, GA, Wisc, Nevada et al. We will prove it all.”
Although the states have yet to officially certify final election results, in keeping with normal practice, presidential candidates traditionally have recognised the independent analysis of major US media and companies such as Edison Research, which called the election for Mr Biden.
He continued: “Even those many millions of Americans who loathe and detest you will have a grudging respect if you now concede gracefully, congratulate your opponent for his win, organise a smooth peaceful transition of power and turn up at his inauguration as Hillary turned up at yours. “That could help you if you decide to run again in 2024, or if one of your children does.
“The Trump brand remains stronger than ever despite all the predictions that it would be irrevocably repudiated in this election.
“But if you refuse to concede, continue to falsely scream that the election was stolen, and pathetically snub President-Elect Biden, then you won’t just go down as a loser, you’ll go down as the sorest loser in US presidential history.”