The ECB, however, cleared all-rounder Liam Livingstone, who is coming back after recovering from his knee and ankle injuries, to play in the IPL.
Inglis was injured when a club snapped in his hand during a "normal golf swing" while on a morning round with some of his teammates.
Out-of-form opener Jason Roy was dropped from the squad for the tournament in Australia.
The all-format player has potentially sustained a right knee injury.
Root went on to smash a masterly 142 not out, his 28th Test century, forging a mammoth 269-run partnership for the unbroken fourth wicket with Bairstow, who remained unbeaten on 114.
Bairstow said the verbal exchange was part and parcel of the game.
India finished day three at 125/3 after the hosts were bowled out for 284 earlier on the day.
Bairstow's gutsy 103 not out helped his side to 258/7 in reply to Australia's 416/8 declared, a position that looked highly unlikely when England limped battered and bruised to lunch on 36/4.
Aiden Markram has been named Malan's replacement in the Punjab Kings squad.
Rahul wasn't happy with the decision, pointing towards the back pad and shaking his head while walking back to the pavilion.