Team management said there were "clinical signs of an injury to the left Achilles tendon".
The Roma defender will be a big miss for Italy in the remainder of the European Championship after enjoying a fantastic tournament.
Some media reports have suggested the 28-year-old could be out of action for several months.
"The first important thing is that we're sorry for 'Spina'," said Italy coach Roberto Mancini. "He didn't deserve this injury.
"He was playing extraordinarily well. He was even one of the best in the Euro."
Spinazzola pulled up after chasing down a long ball, gesturing to the bench he had to be substituted.
He was stretchered from the field in the 79th minute in tears.
"We'll do everything we can to make it to the final for him," said Lorenzo Insigne, who scored what proved to be the winner in Munich.
Italy, chasing a first European title since 1968, will take on Spain in the first semi-final at Wembley on Tuesday.