The 23-year-old Bairstow has been replaced in the squad by Joe Root, who made his England debut in the fourth Test against India in Nagpur.
Bairstow was a part of the England's Test and Twenty20 squads against India but had to fly back home prior to the second and final T20 match before Christmas, for the same reason.
England will play a five-match ODI series against India, commencing in Rajkot on January 11.
He will remain at home with his family but is expected to join the English team for the T20 and ODI series against New Zealand in February.