Rakitic was substituted during the second half of Croatia's dramatic 3-2 victory at home to Spain on Thursday, with Tin Jedvaj hitting a 93rd-minute winner.
That result set up a decider against England at Wembley on Sunday, but Rakitic will be unable to play due to his injury.
"I asked for the change because I had discomfort," Rakitic said after the Spain win.
Croatia will qualify for the finals of the inaugural Nations League if they beat England in a repeat of the World Cup semi-final, but will be relegated to League B if they lose.
Dalic confirmed the injury news at Zagreb airport, with Barcelona midfielder Rakitic set to return to his club for treatment.
"He will go to Barcelona because there is an injury," Dalic told reporters. "You know my practice not to risk [players] if there is anything suspicious.
"We want healthy and ready players, Ivan Rakitic at the moment is not one hundred per cent ready for the game so unfortunately we go without him.
"We've got [Nikola] Vlasic, [Mateo] Kovacic, [Josip] Brekalo, we have [Milan] Badelj, we always have solutions. We'll look for another solution and that will be fine, I hope."
Rakitic's injury is also bad news for Barcelona, although he is unavailable for their next LaLiga game, away to title rivals Atletico Madrid a week on Saturday.
He was sent off in Barca's last match, a 4-3 reverse at home to Real Betis, so he will be serving a suspension.