He returned to Aviva Premiership action in March but lasted just five games before he suffered a reoccurrence.
"I have spoken to [England head coach] Stuart Lancaster," nine-time capped Wood told BBC Radio Northampton.
"Having not played any rugby in the tail end of the season and not being fit to put my best foot forward for the Saints I can't go on that tour with any run of form or fitness.
"So I don't think that's a wise move at this stage and the lads have been going very well without me so I wish them the best of luck."
Wood, a potential Lion next year, was a front-runner to be named as England captain after Lewis Moody stepped down following the World Cup.
But he has not featured for the Red Rose since the group stage 67-3 victory over Romania at the global tournament last year.
England face the Lions last opponents during a three-Test tour in June.