Vunipola had been nursing the injury throughout the Aviva Premiership while playing for Saracens.
England backrower James Haskell will replace Vunipola and will join the camp following the Premiership final this weekend.
Vunipola, who was born in Australia, has played 34 tests for England and was expected to be the starting number eight for the Lions in their three-test series against the All Blacks.
Lions coach Warren Gatland said he was disappointed for Vunipola, who had been looking forward to joining the squad
"He has been carrying an injury and feels he wouldn't be able to contribute fully to the Tour and needs further medical treatment. We really appreciate his honesty in making this decision and wish him luck with his recovery.
"We have called up James to the squad, we wish him all the best next weekend and look forward to welcoming him into camp before we depart."