Indoor gatherings are the greatest risk posed by the major tournament, Prof Bauld said, as she implored fans to “just do it outdoors”.
It comes after the R number in Scotland – the average number of people someone with Covid passes the virus on to – rose again, to between 1.2 and 1.4.
Prof Bauld, professor of public health at Edinburgh University, said the situation in Scotland was “fragile”.
“I am very concerned about the next four weeks," she said, adding that cases could double in the next week.
Last week case numbers increased by 35 per cent.
Prof Bauld said while she doubts another national lockdown would be introduced, Scotland may see a return to local restrictions.
She is also “pretty confident” the easing due to take place on June 28 will not happen.
“I don't think we're going to see another national lockdown quickly – or hopefully ever,” she said.
"But I think the next little while is a concern. The good weather will help, but I think we're at quite a fragile situation.”
Prof Bauld said she would have liked to see mandatory testing at Glasgow’s fan zones, where thousands of people will gather to watch matches.
But while these are outside, indoor gatherings are the biggest risk, she said.
“The main risk is more people from different households mixing indoors in enclosed spaces, that's the key thing,” she said.
“I can't emphasise enough that this virus is airborne. If you're standing in a room, even a little bit away from somebody else, with poor ventilation, and you've got one infected person in the room, if you're in there for a while having a few drinks etc, it's going to pass around the room just like second-hand smoke.”
She advised people to “just do it outdoors” if possible.
Prof Bauld added: “If you are doing it indoors, either try and watch the match maybe with a friend in the pub, which you can do, or if you're going to somebody else's house keep the number of people down, don't spend too long there, and open the doors and windows – that’s the single most important thing.”
Prof Bauld said those gathering at house parties were likely to be younger – up to people in their 30s – many of whom have not had any vaccine, and very few of whom have had both doses.
“That means that they can develop the disease, and if many of them are infected then some of them may end up in hospital, which we don't want,” she said.
“Secondly they may develop long Covid, which is looking like it's around one in ten people, and that can have debilitating implications for their health going forward.
“Thirdly if they're carrying it and then moving around with other family members who have already had the vaccine, but are not 100 per cent protected – remember we are seeing breakthrough cases in a small number of older people – then they're putting them at risk, although clearly far less risk than used to be the case in the past.”