Although Richardson suffered the injury while batting in the nets in Hyderabad last week, he trained on the eve of the second and final T20 International on Wednesday.
"Kane reported some left side pain at training prior to the opening T20I in Vizag," said Australia team physio David Beakley in a statement.
"Unfortunately he has not recovered well enough in order to play a further part in the tour. Kane will return home to continue his rehabilitation, and we (will) monitor his progress over the coming weeks."
Richardson's absence is a significant blow for the visitors who are already missing the services of injured pacers Mitchell Starc and Josh Hazlewood.
The 28-year-old from South Australia was in top form in the recently concluded Big Bash League, ending as the highest wicket-taker for eventual champions Melbourne Renegades.
The injury to Richardson gives an opportunity to Tye, who last played for Australia in the drawn home T20I series against India in November.
The 32-year-old from Western Australia has plenty of experience of playing in India, having been part of IPL teams Chennai Super Kings, Kings XI Punjab and now defunct Gujarat Lions.
Tye has played 26 T20 International and seven ODIs, taking 37 and 12 wickets respectively.
Australia, who have never been beaten in a T20I series, lead the two-match series 1-0. The T20s will be followed by five ODIs starting in Hyderabad on March 2.