Cohen, a World Cup winner in 2003, was included in Brian Ashton's 30-man squad for England's two-Test tour on Tuesday.
But the Northampton winger has decided to put his international career on hold for the year.
Ashton said: "Ben was very pleased to hear the news that he would be touring with England but he has reflected on his selection and his current personal circumstances.
"His wife is expecting twins later in the year so I understand his reasons for withdrawing from the tour and from contention for the Rugby World Cup squad to spend time with her."
Ashton will announce Cohen's replacement in the England squad on Tuesday.