The 29-year-old's condition has been "stabilised", according to tournament organisers Uefa.
The Group B game was suspended before half-time after the Inter Milan player fell to the ground.
Referee Anthony Taylor immediately called for medical support and Eriksen received treatment on the field.
Updates after Denmark v Finland postponed
Eriksen received CPR before the players were taken off the pitch and the match was postponed.
Uefa confirmed the match had been suspended because of a medical emergency.
"Following the medical emergency involving Denmark's player Christian Eriksen, a crisis meeting has taken place with both teams and match officials and further information will be communicated," it said.
"The player has been transferred to the hospital and has been stabilised."
The incident happened as a throw-in was played towards him near the end of the first half.
His distressed team-mates surrounded him and the shocked supporters in Copenhagen were visibly upset as the player was treated.
Saturday's other Group B match between Belgium and Russia, scheduled for 20:00 BST, is set to take place as scheduled.
Eriksen's club, Inter, tweeted: "Forza Chris, all of our thoughts are with you."
Former team Tottenham said: "Get well soon, Christian. The whole Spurs family is with you."
Last month, Eriksen helped Inter Milan clinch their first Italian title for 11 years in his first full season with the club, after joining from Spurs for £16.9m in January 2020.
The Ajax youth player spent seven years with Tottenham, scoring 69 goals in 305 appearances.
More to follow....
Source: BBC Sport
Last modified on Saturday, 12 June 2021