Bairstow's unbeaten 108 off 48 balls, with eight fours and nine sixes, see Punjab secure the win with eight balls to spare
The left-hander was not named in the 13-man squad announced late on Thursday, with Guyana left-hander Leon Johnson called up as a replacement, reports CMC.
A West Indies Cricket Board (WICB) release said Gayle had been unavailable for selection.
“He has been given permission to deal with a personal matter,” the release said.
Gayle is one of the squad’s most experienced batsmen with 7,214 runs in 103 Tests.
He scored 64 and nine not out in the first Test against Bangladesh which the West Indies won by 10 wickets.
Johnson, meanwhile, is yet to play a Test despite being on the verge of the squad in recent times.
His inclusion is the only change to the squad from the first Test.