The 28-year-old lasted just ten minutes of Wednesday's 1-1 draw with Syria, before being stretchered off.
"Right now I want the injury to heal as quickly as possible," Kagawa said in a statement released by the Japan Football Association, confirming his withdrawal from the squad to face Iraq in FIFA World Cup 2018 qualifying on 13 June. "I hope the players will fight for victory."
Kagawa, who made 21 Bundesliga appearances in 2016/17, is expected to be out of action for up to four weeks.
DFB Cup winners Dortmund start pre-season training on 7 July.
Watch: Shinji Kagawa on Dortmund's global fan base