The midfielder dropped to the ground at the Parken Stadium in Copenhagen shortly before half-time, leaving players from both teams in clear distress.
Politicians and celebrities shared messages of support for the family of the former Tottenham player, who is reported to be awake and recovering in hospital.
In a statement, the BBC said: "We apologise to anyone who was upset by the images broadcast.
"In-stadium coverage is controlled by Uefa as the host broadcaster, and as soon as the match was suspended, we took our coverage off air as quickly as possible."
The first round match between the two nations was suspended but resumed at 7.30pm "following the request made by players of both teams", according to Uefa.
Denmark tweeted an update which read: "Christian Eriksen is awake and is (set) for further examinations at Rigshospitalet (hospital)."
UK Culture Secretary Oliver Dowden was among those who posted online about the "awful scenes" witnessed at the match in Copenhagen, as Eriksen's teammates surrounded him to protect his privacy.
Former Spurs striker Gary Lineker, who was in the BBC studio for the match, also apologised for the coverage but said they "were the host pictures and out of our control".
He added: "They should have stayed on a wide of the stadium. Apologies."
Writing on Twitter shortly after the incident, Mr Dowden said: "Awful scenes at Denmark v Finland Euros game. Thoughts with Christian Eriksen and his family."
Fabrice Muamba, the former Bolton midfielder who collapsed after suffering a cardiac arrest during a televised FA Cup game in 2012, tweeted "Please God" following the incident.
Other players posting similar tweets for the Denmark midfielder included former Liverpool defender Jamie Carragher and Manchester United's former right-back Gary Neville as well as current England and Aston Villa forward Jack Grealish.
David Lammy, the Labour MP for Tottenham and shadow justice secretary, tweeted: "Prayers and thoughts for Danish and Spurs Legend Christian Eriksen.
"We are all rooting for you brother."
London Mayor Sadiq Khan tweeted: "Deeply upsetting scenes in Copenhagen.
"My thoughts right now are with Christian Eriksen, his family, and everyone in Denmark and around the world praying for his recovery."