The 24-year-old fast bowler has been told that he must now require surgery.
Milne was signed in January to cover the second half of the T20 Blast qualifying group campaign, while Pakistan's Wahab Riaz was then brought in to cover the first seven matches.
But Riaz now joins up with Pakistan for the Test series with England.
Essex, second bottom of the southern group with just two points from five games, now hope to find a stand-in at short notice.
"We have a list of replacements that we are working hard to bring in," said Essex head coach Chris Silverwood. "Hopefully we can treat the fans to an exciting arrival in the coming weeks.
"It is disappointing for ourselves and Adam. We were excited by his arrival. But, unfortunately, he requires surgery on an elbow injury he picked up previously."
Although struggling in the T20, Essex remain top of Division Two in the County Championship and also head their One-Day Cup group after four games.
Adam Milne has not appeared since being ruled out with a hamstring injury after his Indian Premier League debut for Royal Challengers Bangalore in mid-April.
Playing in the same team as Chris Gayle, Virat Kohli and AB de Villiers, he went for 43 runs from his four overs in a 45-run win over Sunrisers.
Milne is the third New Zealander forced to return home early from T20 duty, following in the wake of Worcestershire's Mitchell Santner (broken finger) and Birmingham Bears wicketkeeper Luke Ronchi, who has been selected for the Kiwis' forthcoming tour of Zimbabwe and South Africa.
New Zealand international Adam Milne is to miss his scheduled seven-match T20 Blast stint with Essex following an elbow injury.