“Nothing can justify what was seen in Leipzig,” Justice Minister Christine Lambrecht said in a statement posted to Twitter.
“The mockery of science and the far-right incitement we've seen are appalling,” she added, condemning attacks on police and the press.
More than 20,000 people, most of them unmasked, gathered in the Saxon city on Saturday, and many refused a police order to disperse after ignoring requests to wear face coverings and keep a distance of 1.5 metres between participants.
Rather than leave, demonstrators set off on a march down one of the city's major streets, attacking police and journalists and throwing objects including fireworks, police said.
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Foreign Minister Heiko Maas said those “endangering fellow people, attacking police and journalists, spreading extreme-right hate and setting alight barricades at counter-demonstrations have left the protection” for protest guaranteed under Germany's constitution.
Chancellor Angela Merkel's spokesman Steffen Seibert did not comment directly on the violence, but recalled that demonstrators are bound by infection control measures.
Violence continued into the evening in Leipzig, and a police spokesman told AFP there had been “arrests”.
Germany has in recent days hit new records for coronavirus infections above 23,000 in a single day, while the number of dead since the pandemic began has mounted to 11,226.
Unlike some European neighbours like France, the country has not ordered a nationwide lockdown.
But the discipline most Germans have shown so far in following government-ordered measures has been tinged with discontent in some quarters.