Vettori, New Zealand's most capped Test cricketer, suffered a right abductor strain during the first Test against West Indies last month.
While the team under Ross Taylor are certain to miss Vettori's experience in India's spin-friendly conditions, they would expect Patel, a 32-year-old off-spinner of Indian origin, to fill the void.
"Unfortunately Daniel Vettori won't recover in time for the Test series but his injury has created an opportunity for Jeetan Patel to come back into the side," national selection manager Kim Littlejohn said.
"Jeetan has been in great form for Warwickshire and deserves his opportunity."
Batsman Dean Brownlie made way for all-rounder James Franklin in the only other change to the squad that recently lost 2-0 in the Test series against West Indies.
"The tour of India will be a great challenge and experience for this squad," Littlejohn added. "James Franklin has been brought into the squad to bolster the batting.
"He brings a lot of experience and knowledge of playing in Indian conditions which will be invaluable.
"Dean Brownlie misses out on selection. We continue to see him as player of the future but at this stage we feel he needs to works on parts of his game."
Hyderabad hosts the first Test from August 23, while the second match is scheduled in Bangalore from August 31.
Squad: Ross Taylor (captain), Trent Boult, Doug Bracewell, Daniel Flynn, James Franklin, Martin Guptill, Chris Martin, Brendon McCullum, Tarun Nethula, Jeetan Patel, Tim Southee, Kruger van Wyk, Neil Wagner, BJ Watling, Kane Williamson.