The violence in the capital Belgrade was triggered when a crowd stormed parliament in protest at plans to reimpose a lockdown following a surge in COVID-19 cases.
"There are no free beds in our hospitals, we will open new hospitals," Vucic said in an address to the nation. He also accused far-right organizations and unspecified regional "intelligence officials" of staging riots to "undermine Serbia's position".
Serbian opposition parties and activists announced more protests in Belgrade later in the day.
Footage showed police kicking and beating people with truncheons, while protesters pelted officers with stones and bottles, after thousands chanting for the resignation of Vucic gathered outside the parliament building on Tuesday.
Although he had on Tuesday said a new lockdown would be imposed in Belgrade at the weekend, Vucic said on Wednesday he had ultimately advised the government and health authorities not to introduce it. The government would announce a new set of sanitary measures on Thursday, he said.
Earlier, police director Vladimir Rebic said that 43 police and 17 protesters were hurt and 23 protesters had been arrested.
The government's critics say its decisions to allow soccer matches, religious festivities, parties and private gatherings to resume, and parliamentary elections to go ahead on June 21, are to blame for the spike in infections.
Serbia, a country of 7 million, has reported 17,076 COVID-19 cases and 341 deaths. Health authorities say hospitals are running at full capacity and staff are exhausted.
The government says a lack of sanitary discipline among the population is the reason for the rise in cases.Opposition parties, many of which boycotted an election that Vucic's Progressive Party won by a landslide, say he is using the pandemic and lockdowns to strengthen what they call his autocratic rule, accusations that he denies.