Sam Billings will be unavailable for game two of the series on Friday, with a decision on his availability for Sunday’s final match to be taken in due course.
Jos Buttler will captain England in Morgan’s absence and Liam Livingstone will make his ODI debut.
Dawid Malan, with the tour party as cover, has been drafted into the squad and is available for selection.
Morgan sustained a split webbing between his thumb and index finger in Tuesday’s series opener, an injury that required four stitches.
The captain went through a fielding drill at the start of training at the MCA Stadium on Thursday and afterwards declared himself unfit.
Morgan said: “I had the hand re-dressed before the start of training today but it quickly became apparent that I found myself protecting the injury and getting into the wrong positions to catch the ball.
“There’s nowhere to hide on the field in international cricket these days, especially in limited-over formats so it didn’t require any great thought to make myself unavailable.
“It was a freak injury and it’s extremely frustrating but there’s nothing I can do about it. It’s now just a case of letting the cut heal. I have every confidence in Jos and the rest of the squad coping without me.”
Billings did not train on Thursday after he suffered a bruised left collar bone while fielding in Tuesday’s opening match, and the injury remained sore on Thursday.