Mr Biden believes it is a “good thing” Mr Trump will not be attending his inauguration on January 20. Five people died in riots at the US Capitol on Wednesday, including police officer Brian Sicknick, law enforcement said in a statement.
At least 60 people have been arrested after demonstrators broke into the Capitol in a bid to disrupt the certification of the Democrat’s November election victory.
Mr Biden said: “He is not fit to serve.
“He is one of the most incompetent presidents in the history of the United States of America.
“I was told on the way over here that he wasn’t going to show up at the inauguration – one of the few things he and I have ever agreed on.”
Mr Trump tweeted: “To all of those who have asked, I will not be going to the Inauguration on January 20.”
Traditional inauguration procedures see incoming and outgoing presidents ride to the US Capitol together ahead of the ceremony.
Mr Biden said Vice President Mike Pence would be welcome at his inauguration.
Mr Pence defied Mr Trump’s pressure to block certification of the election result on Wednesday.
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He told those in attendance: “We will never give up, we will never concede.”
At 1pm Eastern Time, pro-Trump protesters breached barriers around the Capitol Building, overwhelming officers in full riot gear.
By 3pm, officials had been evacuated and an armed standoff ensued.
Protester Richard Barnett was photographed sitting in Ms Pelosi’s office chair and has now been arrested, federal officials said.
The Republican said he is focused on ensuring “a smooth, orderly and seamless transition of power”.
He added: “The demonstrators who infiltrated the Capitol have defiled the seat of American democracy.”
“To those who engaged in the acts of violence and destruction, you do not represent our country, and to those who broke the law, you will pay.
“We have just been through an intense election and emotions are high but now tempers must be cooled and calm restored.”