Demonstrators at Evangelismos hospital in central Athens held up signs demanding more hiring, virus testing and hospital equipment and were dispersed by the police. A similar protest was held at the main hospital in Larissa in central Greece.
Greek authorities had earlier quarantined the Ritsona camp in central Greece after 20 asylum seekers tested positive for coronavirus.
Refugees at the Moria camp on the Greek island of Lesvos are constantly in danger of contracting Covid-19 as hygiene and social distancing seem impossible.
Overcrowding in camps and poor hygiene means refugees face a heightened threat from the coronavirus.
For the displaced living in crowded camps, social distancing and regular hand-washing is not an option
The flame, carried in a burning canister, touched down amid growing doubts if the Tokyo Games can open as scheduled on July 24 because of the coronavirus pandemic.
Turkey had earlier declared that it was closing down its six land and sea border crossings with Greece and Bulgaria in an effort to contain the spread of the novel coronavirus.
The IOC has remained committed to staging the Tokyo Games as planned from July 24-August 9, saying on Tuesday after a meeting with international sports federations that measures against the virus were delivering results.
"The temporary shutdown of cross-border movement within the EU will inevitably impact when and how vulnerable children in Greece are relocated to Germany, Finland and other states," a senior UNICEF official said.
The fire burned through two containers used as living quarters by people in the Moria camp, as well as some tents, a spokesman said.
Weeks after making their way to the Greek border region, refugees and migrants are holding out despite Greek attempts to push them back.
Turkey announced in February it would no longer stop refugees and migrants from reaching Europe via land, while banning boat crossings as too dangerous, accusing the EU of failing to keep promises under the 2016 refugee deal.
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