Cook, 32, had been made available by the England and Wales Cricket Board (ECB) for the county's first three Division One matches.
The England and Wales Cricket Board say the injury is "left hip soreness" and he is to have an injection.
"I've been fairly injury-free in my career so far. It's frustrating to get a niggle now." Cook told BBC Look East.
Essex, champions of Division Two in 2016, begin the new season against Lancashire on Friday.
It means Cook will miss a chance to face England team-mate James Anderson at Chelmsford.
"It's a shame it's not happening, but it is what it is. I'm just disappointed to miss the first game back in Division One of the County Championship." Cook added.
"But there is a bigger picture to look at, and the benefits of risking one game don't outweigh getting it fixed and getting it right."
Cook resigned as England captain in February after a record 59 Tests in charge, with Joe Root taking up the role for this summer's home series against South Africa and the West Indies.
A statement on the ECB website said: "Cook has been receiving treatment for the problem for several weeks.
"A scan this week has ruled out serious injury, but he will undertake an injection in the upcoming days to help settle his symptoms."
Former England Test captain Alastair Cook will miss Essex's opening County Championship game with a hip injury.