The England and Saracens number eight had been trying to manage the injury in recent weeks, but he has now confirmed that it will require further treatment which will rule him out of the squad.
Vunipola, who had been expected to be the Lions' starting number eight for the first Test against the All Blacks, will be replaced in the squad by Wasps back-row James Haskell.
"It is really disappointing for Billy and we know how much he was 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," read a statement from head coach Warren Gatland.
"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."
Haskell will join up with the rest of the squad after the Aviva Premiership final on May 28.