Boris Johnson has urged the country to “work peacefully, lawfully” to defeat racism and discrimination.
Writing in The Voice, the PM said the government could not ignore the anger and “undeniable feeling of injustice” sparked by George Floyd’s killing.
But he said the cause was at risk of being “undermined” by a minority of those attacking police and property.
The UK, he said, had made “huge strides” in tackling racism in recent decades but more had to be done.
The killing of George Floyd while in police custody in the United States 10 days ago triggered an international outcry and sparked days of mass protests in cities across the UK.
While the demonstrations have been largely peaceful, they turned violent in London on Saturday when police officers were confronted with flares, and a statue of wartime leader Winston Churchill was vandalised.
In Bristol, demonstrators pulled down a statue of the slave trader Edward Colston and threw it into the River Avon.
Home Secretary Priti Patel condemned a “lawless minority of protesters” who “regrettably turned to violence”, adding: “This hooliganism is utterly indefensible.”
In an article for the Voice, the prime minister said Mr Floyd’s death had “awakened an anger and a widespread and incontrovertible, undeniable feeling of injustice, a feeling that people from black and minority ethnic groups do face discrimination: in education, in employment, in the application of the criminal law”.
“We simply cannot ignore the depth of emotion that has been triggered by that spectacle, of a black man losing his life at the hands of the police,” he wrote.
“We who lead and who govern simply can’t ignore those feelings because in too many cases, I am afraid, they will be founded on a cold reality.”
‘More to do’
While he said he believed the UK was a much less racist society than it was 40 years ago, the PM said much more needed to be done to ensure everyone was treated equally under the law.
“We must also frankly acknowledge that there is so much more to do – in eradicating prejudice, and creating opportunity, and the government I lead is committed to that effort.”
While supporting the right to peaceful protest, the PM warned that further mass demonstrations endangered the UK’s efforts to fight coronavirus.
And he said he could not condone those who “hijacked” peaceful protests by breaking the law or desecrating public monuments, saying they damaged the legitimate cause that people were fighting for.
“We have a democracy in this country. If you want to change the urban landscape, you can stand for election, or vote for someone who will.
“And so I must say clearly that those who attack public property or the police – who injure the police officers who are trying to keep us all safe – those people will face the full force of the law.
“So let’s work peacefully, lawfully, to defeat racism and discrimination wherever we find it, and let us continue to work together across all the communities of this country, as we put Britain back on its feet.”