Edinburgh University’s Dr Christine Tait-Burkard said allowing fans to gather was a “risk”, and urged them to test themselves before attending.
The government said testing beforehand was encouraged but not mandatory and measures were under ongoing review.
Euro 2020 kicks off on 11 June, with Glasgow one of 11 host cities.
The testing call comes as restrictions are due to ease in Glasgow, with people allowed drink inside pubs and restaurants for the first time since November as the city moves to level two on Saturday.
Up to 6,000 ticketholders a day will be able to watch games on big screens in the fan zones at Glasgow Green over the 31 days of the tournament.
A total of 3,000 people will be able to attend an afternoon session and another 3,000 an evening session in the “largely seated” venue which will be subject to social distancing rules.
About 12,000 fans will also gather in Hampden Park for four matches this month, taking the national stadium to 25% capacity.
They include the Scotland v Czech Republic game on 14 June and the Scotland v Croatia match on 22 June.
Speaking to BBC Radio’s Good Morning Scotland, Dr Tait-Burkard said that while it was “positive news” that fans would be outdoors, she urged policy-makers to insist on testing for those in attendance.
In the absence of mandatory testing, she encouraged fans who would be attending to order free lateral flow tests online from the government, and to take one before going to a fan zone.
Dr Tait-Burkard added: “It’s nice to see Scotland in the Euros, so it’s obviously something that lots of people want to celebrate and would like to celebrate with others.”
However, she added: “With no testing in place, it comes with a risk because we have people who are very enthusiastic and – especially if the teams are doing well – we know that the distancing rules will go out the window very quickly. So be careful.
“Otherwise, I think the government just has to rethink the testing for this one, because we have seen… less transmission in well-tested gatherings.”
During Friday’s Scottish government Covid briefing, First Minister Nicola Sturgeon said it was “not about thinking/rethinking” the fan zones, adding that “these things are kept under ongoing review”.
She said she would consider the latest data over the weekend to see whether that changes the thinking about the mitigations in place around the fan zone.
A lot of effort had been put into making the fan zone “as safe as it possibly could be”, but she stressed that “in a global pandemic of an infectious virus, nothing is 100% safe”.
Ms Sturgeon added that nowhere in the UK was testing mandatory, and said there were “ethical issues” about making testing or any other health intervention mandatory.
She stressed that everyone was asked to test themselves twice a week with lateral flow devices supplied by the government, which can be delivered to your home for free.
“My appeal to people is test yourself [twice a week] routinely and particularly before you go to any event or anywhere where you’re coming into contact with other people,” she added.
Scotland’s national clinical director Jason Leitch added that pre-pandemic Glasgow Green had a capacity of 81,000 people. He said it had been reduced to 3,000 because of the pandemic and added that for some matches it would not meet its capacity.
A Scottish government spokesman added: “The fan zone is ticketed and people’s contact details will be recorded to allow for any track and trace.
“We are encouraging everyone to take up the offer of twice-weekly rapid lateral flow testing. Anyone who has a positive result or any Covid symptoms should not attend the event and should get a PCR test.”
Plans for the fan zone were drawn up by Glasgow Life – the charity which provides culture and leisure facilities for Glasgow City Council – and approved by the Scottish government.
A Glasgow Life spokeswoman said: “It is not a requirement for ticket holders to show a negative result from a Covid-19 test to gain access to the fan zone or the stadium on match days.
“However, in the interest of public health, we would encourage all fans to take a test prior to attending to help stop the spread of the virus as the country moves out of lockdown.
“Event plans will be continually assessed by partners against epidemiological conditions and Covid measures will continue to be monitored in the run-up to and throughout the tournament to ensure they remain appropriate.”
Hospitality bosses have also voiced concern over the impact on local businesses. More than 50 premises put in an objection to the licensing board about the zone.
The Scottish Hospitality Group previously said it was “staggering” that the fan zone would be allowed to go ahead while the industry was forced to operate under strict rules and restrictions.
The Scottish Licensed Trade Association said it feared the fan zone could lead to further Covid outbreaks and a fresh lockdown, “forcing licensed premises to close again when they have only just managed to start reopening”.