Boris Johnson has said anti-racism demonstrations were "subverted by thuggery" after some protesters clashed with police over the weekend.
The prime minister said people had the right to protest peacefully but those who clashed with police were "a betrayal of the cause they purpose to serve" - and would be held to account.
Thousands of people took part in demonstrations across the UK over the death of George Floyd in the US at the hands of a white police officer.
Some 12 people were arrested and eight police were injured during protests in London on Sunday, which saw some clashes between demonstrators and officers.
The statue of Winston Churchill in Parliament Square was also defaced, while in Bristol, a statue of slave trader Edward Colston was torn down during the protests.
The prime minister posted on Twitter: "People have a right to protest peacefully and while observing social distancing but they have no right to attack the police.
"These demonstrations have been subverted by thuggery - and they are a betrayal of the cause they purport to serve.
"Those responsible will be held to account."
Avon and Somerset Police said they had launched an investigation into the toppling of the bronze statue of the 17th Century slave trader on Sunday.
Kit Malthouse, the policing minister, condemned the move as "an act of criminal damage" and said prosecutions could follow for those involved.
He told BBC Radio 4's Today programme: "The argument about whether he should be there memorialised in a statue is separate from whether we allow groups of people just to go about doing as they wish.
"I know it’s been a source of significant debate in the city for some time and it is obviously unfortunate that a proper democratic decision hadn't been reached about what to do with the statue.
"But we can’t condone these kinds of acts of criminal damage and vandalism and it shouldn’t have happened."
Mr Malthouse went on: "We have many statues across the United Kingdom, with figures from history, who have good and bad attached to them we can’t just have a kind of arbitrary rule about who shouldn’t shouldn’t go up."
The minister said there would need to be a "post mortem" into the policing of Black Lives Matter demonstrations but said it would not have been practical to arrest people for attending protests to prevent the spread of coronavirus.
Mr Malthouse said the impact of the protests on infection rates would become clear in the coming weeks after scientists predicted that one in 600 people have the virus.
He said: "We will all have to hold our breath over the next couple of weeks to see exactly what the impact has been. It is very unfortunate.
"As we have said right from the start, we urge people not to participate on public health grounds, notwithstanding the obvious strength of feeling that they have."
The Mayor of Bristol, Marvin Rees, said it was important for politicians to listen to those who found the Colston statue to represent an "affront to humanity".
He said: "As an elected politician, obviously I cannot condone the damage and I am very concerned about the implications of a mass gathering on the possibility of a second Covid wave.
"But I am of Jamaican heritage and I cannot pretend that I have any real sense of loss for the statue and I cannot pretend it was anything other than a personal affront to me to have it in the middle of Bristol, the city in which I grew up."
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