US president-elect Joe Biden has demanded that Donald Trump goes on national television "to fulfil his oath" and call for peace.
It comes as Republican Vermont governor Phil Scott said on Twitter that the "fabric of our democracy and the principles of our republic are under attack by the President".
"President Trump should resign or be removed from office by his Cabinet, or by the Congress," he added.
Reacting to protests at the US Capitol building in Washington DC, Mr Biden said he was shocked and saddened and insisted members of the "mob do not represent who we are" and described them as "a small number of extremists dedicated to lawlessness".
"This is not dissent, it's disorder, chaos and borders on sedition and it must end now," he added.
"Pull back and allow the work of democracy to go forward. The world is watching."
He then called on Donald Trump to step up and urge the protesters to pull back.
"I call on this mob to pull back and allow the work of democracy to go forward. You heard me say before in a different context - the words of a president matter, no matter how good or bad that president is.
"At their best, the words of a president can inspire. At their worst, they can incite.
"Therefore, I call on President Trump to go on national television now to fulfil his oath and defend the Constitution and demand an end to this siege."
Shortly after, the president released a recorded statement on Twitter.
"I know your pain, I know you're hurt," he said.
"We had an election stolen from us...but you have to go home now, we have to have peace.
"This was a fraudulent election but we can't play into the hands of these people but we have to have peace."
Vice President Mike Pence - who Mr Trump had earlier accused of lacking courage after he said he would not intervene in the election result - also tweeted about the unrest.
"The violence and destruction taking place at the US Capitol Must Stop and it Must Stop Now. Anyone involved must respect Law Enforcement officers and immediately leave the building," he said.
"Peaceful protest is the right of every American but this attack on our Capitol will not be tolerated and those involved will be prosecuted to the full extent of the law."