The ECB announced that Liam Livingstone would make his debut in the second ODI on Friday and that Dawid Malan, who was in the squad as a cover, would be available for selection in the remaining two games. Jos Buttler will lead the England team in Morgan's absence.
Morgan pronounced himself unfit after going through a fielding drill on Thursday at the MCA Stadium in Pune, the venue for all three ODIs.
"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," ECB's website quoted Morgan as saying.
"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."
Though Livingstone is certain to make his debut, Malan may replace Morgan at XI in the second ODI as a like-for-like replacement. After losing the first game by 66 runs, England will try to save the series on Friday.
The series' final match will take place on March 28.