Mr Trump's remarks – posted on Twitter on Thursday evening following a suspension – were widely seen as his first public acknowledgement of electoral defeat.
It comes after he was widely condemned for provoking violence from a mob of loyal supporters who stormed the US Capitol in an attempt to overturn the election and prevent Biden taking to the White House.
Protesters then clashed with police, one woman died after being shot, and three other people died after suffering “medical emergencies”.
Mr Trump spoke as top Democrats called for him to be removed from office.
He returned to Twitter following a 12-hour freeze of his account after the social media company said his tweets could stoke violence.
In a marked change of tone, he condemned those who stormed the US Capitol, describing the scenes as “mayhem” and calling it a “heinous attack”.
He said in the clip: “Now Congress has certified the results a new administration will be inaugurated on January 20th.
“My focus now turns to ensuring a smooth, orderly and seamless transition of power. This moment calls for healing and reconciliation."
On Thursday, Scotland’s Justice Minister called on Home Secretary Priti Patel to deny Mr Trump entry to the UK once he leaves office.
Humza Yousaf tweeted: “Once he leaves office if Trump tries to come to UK the Home Secretary should give serious consideration to denying him entry, she has the power if an applicant’s presence is not conducive to the public good.
“Trump’s default is to stir up racial tension and yesterday he incited a violent mob.”
Nicola Sturgeon also said that the scenes in Washington were “shocking”, but not “surprising”.
The First Minister told ITV’s Good Morning Britain: “On one level I think what happened last night, what we witnessed last night, is not that surprising.
“In some senses Donald Trump’s presidency has been moving towards this moment almost from the moment it started, but that doesn’t make it any less shocking.
“What we witnessed weren’t just scenes of horrible breaches of law and order, people taking over the seat of democracy, we actually witnessed the president of the United States inciting insurrection in his own country and I think for many people it will take some time to get our heads round that.
“Thankfully there’s only a matter of days of his presidency left. We heard Joe Biden last night remind us what a real leader, a real democratic leader, should sound like.
“This has been a dark period in America’s history, there’s no doubt about it, and I would imagine many people in that country and across the world are looking forward to it coming to an end but clearly there are deep-seeded divides there that the new administration has to tackle and try to heal.”
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