The 28-year-old Yorkshireman suffered the problem during England's disappointing World Cup campaign and was forced to have a spell on the sidelines recovering.
However Gough suffered a recurrence of the injury during Yorkshire's recent championship match with Sussex and must now miss the opening Test..
Headingley officials have consulted England about Gough's problem and have agreed he needs rest and rehabilitation, which could rule him out for up to a month and possibly sideline him for the second Test at Lord's starting on July 22.
Martyn Moxon, Yorkshire's Director of Coaching, said: "Darren has been withdrawn from today's National League game with Essex due to a niggling calf strain which has failed to heal satisfactory.
"After a full examination by the Yorkshire physiotherapist, it has been decided that he requires rest, treatment and rehabiliation."
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