The president-elect laid blame for the riotous scenes in Washington on Wednesday with Trump, saying he was guilty of "trying to use a mob to silence the voices of nearly 160 million Americans" who voted in November.
Biden's election win has now been certified by Congress - who returned after being evacuated from the chamber following the violence - and Trump has committed to an "orderly transition" of power.
Watch ITV News Washington Correspondent Robert Moore's on the fallout from the riots:
Speaking in Delaware on Thursday, president-elect Biden described the group that descended on America's highest government building as "a riotous mob — insurrectionists, domestic terrorists."
The 78-year-old said president Trump had "made his contempt for our democracy, our Constitution, the rule of claw clear in everything he has done" and unleashed an "all-out attack" on the country’s democratic institutions.
But many are calling for actionable repercussions against Trump after his role in inciting the riots, and failure to call off his supporters - calling them, instead, "very special".
Inside the Washington siege - Robert Moore on his report which stunned the world
White House press secretary Kayleigh McEnany condemned Wednesday's violence but stopped short of offering an apology on the president's behalf and took no questions from the press.
"I am here to deliver this message on behalf of the entire White House. Let me be clear, the violence we saw yesterday at our nation's capital was appalling, reprehensible and antithetical to the American way," she said on Thursday evening.
"We condemn it, the president and this administration in the strongest possible terms. It is unacceptable and those that broke the law should be prosecuted to the fullest extent of the law."
Moments after the press conference, reports emerged of US Capitol Police Chief Sund is resigning, effective from January 16, according to NBC News.
House Speaker Nancy Pelosi said President Trump should immediately be removed from office - by invoking the 25th amendment - or Congress may proceed to impeach him.
She said at the Capitol: "The president of the United States incited an armed insurrection against America."
Her voice joins a growing chorus of Democrats - and some Republicans - who want Trump's removal before his term ends on January 20.
A number of high-profile Republicans have also resigned in the wake of the chaos.
The latest among them, US transportation secretary Elaine Chao, said she "cannot set aside" the pro-Trump insurrection that unfolded.
The events were the final straw for social media sites who have long grappled with how to handle Trump, especially on Twitter.
The president has been blocked from using Facebook and Instagram "indefinitely and at least for the next two weeks", founder Mark Zuckerberg has confirmed.
While his Twitter account has been temporarily suspended - and multiple tweets made on Wednesday as events unfolded, deleted.