The Stoke stopper was substituted during the 3-1 defeat to the South Americans in Saturday's clash in Hamburg and will not now travel to Poland and Ukraine.
His place in Morten Olsen's 23-man squad has been taken by Leicester goalkeeper Kasper Schmeichel, the son of Denmark great Peter.
Sorensen said: "Right now I am very disappointed but we need to ensure the proper treatment of the spine. Then I will support my comrades in the national team as best I can from home during the finals."
He later added on Twitter: "Devastated to miss the Euros due to injury. When so much energy and soul has been put into something its a tough one."