The situation is being closely monitored, he told MSPs on Tuesday, ahead of the beginning of the event in Glasgow on Friday.
Up to 3,000 fans will be allowed to gather in two separate sessions each day in an outdoor fan zone at Glasgow Green during the month of games.
They will be seated at picnic-style tables of up to six people, with food and drink, including alcohol, available.
Mr Yousaf’s comments come as it emerged the controversial Glasgow fan zone would cost the taxpayer at least £84,000 just for the big screen televisions.
The fan zone, which is being run by Glasgow Life, a charitable arm of Glasgow City Council that is funded partially through taxpayers’ cash, is set to open on Friday amid calls for additional testing to lower the risk of Covid-19.
The Scottish Government has faced repeated calls for testing at the event, and concern from some quarters about possible spread of the virus in the city, which moved out of level three restrictions on Saturday.
Officials have previously said testing is “not feasible”, and Mr Yousaf confirmed on Tuesday that no plans for testing are in place.
There are “issues” around mandatory testing, he said, including ethical concerns and the fact that some people cannot take a test due to other medical factors or disability.
He also cited digital exclusion as a problem if not all participants can access email to prove a test.
However, he said Glasgow City Council would email all ticket-holders via the ticketing provider to encourage them to take a lateral flow test, which are available to order from the UK Government and perform at home.
Mr Yousaf said the situation “will be continually reviewed in the run-up to and, of course, during the tournament, taking into account the latest scientific and clinical advice and local information that we get on the ground”.
The health secretary said he “understands” concerns over the fan zones, and added that permission for these could still be revoked if deemed necessary.
“The proposal for a fan zone is not about prioritising football over other priorities, it’s about seeking to cater in as safe a way as possible to fans who want to watch the matches,” he said.
“However, I want to give an assurance that these decisions are made carefully, and with full account taken of clinical advice. In the case of the fan zone it will provide an outdoor, highly regulated space for fans to watch the tournament.
“I want to make it clear that the situation with the virus, the application of mitigations, and the actual experience of the event will be monitored on an ongoing basis.
“Any changes considered necessary will be made, up to and including withdrawing permission, should significant concerns arise.”
Mr Yousaf was questioned over testing by Liberal Democrat MSP Alex Cole-Hamilton.
"Asymptomatic testing has been an integral part of trial events across the United Kingdom, including entry to the FA cup final last month,” said Mr Cole-Hamilton.
"Euro 2020 events are being advertised as taking place in a Covid-secure environment, but there is no way on earth of verifying that without knowing the Covid status of every participant.
"Why has the government decided that mandatory testing is not necessary for attendance at the fan zone or the games at Hampden?”
Hospitality groups have criticised the decision to allow the fan zone to go ahead while pubs and bars across Glasgow are forced to close at 10:30pm, potentially when games in the European Championship are still ongoing.
The Scottish Hospitality Group claimed on Monday this could lead to violence between landlords and customers and wasted police time.
Opposition MSPs have also led calls for an robust testing regime at the fan zone to reduce to the risk of Covid-19 outbreaks, but the Scottish Government claimed on Monday testing attendees on arrival would pose a Covid-19 risk.
On Tuesday morning it emerged that Glasgow Life, which runs cultural and sporting events in Glasgow on behalf of Glasgow City Council, will spend at least £84,000 on the big screens and associated staging alone for the fan zone.
The organisation is likely to pay more than this overall, with the value of at least one more contract for cleaning services at the fan zone yet to be confirmed.
On the Public Contracts Scotland website, a public notice of the contract with Adi UK Ltd, a Bolton-based big screen provider, states Glasgow Life has agreed to pay £84,340 for the giant televisions.
It adds: “This contract is to manage the digital content as well as the stage activations, overall presentation and supply of LED screens for the duration of the fan zones for the UEFA Euro 2020 tournament.”
Another contract, for ‘cleansing services’, is yet to have a confirmed supplier.
Its contract listing states Glasgow Life will pay for “a specialist cleaning service to provide daily onsite cleaning services running up to and the duration of the Euro 2020 tournament”.
The contract adds: “The supplier will undertake a full scope of the cleaning operation required, including resource, scheduling, and creation of a full cleaning management plan.”
Glasgow Life has been contacted for comment.
Alongside the fan zone, 12,000 people will be allowed to watch four games in Hampden Park, filling the stadium to 25 per cent capacity.