Root, who was England captain for 64 Tests between 2017 and 2022, was elevated to the leadership role ahead of official Test vice-captain Harry Brook.
Stokes and seamer Atkinson were not considered for selection after a late-night scuffle in a London nightclub following England's victory in the first Test at Lord's with the duo still under investigation by the ECB.
"Given the ongoing investigation, Ben Stokes and Gus Atkinson have not been made available for selection for the Rothesay second Test," the ECB said in a further statement on Wednesday.
Jordan Cox and Jofra Archer have been chosen in a 15-man squad to replace Stokes and Atkinson, but England do not have a like-for-like allrounder to replace Stokes in the XI.
The second Test starts at the Oval on 17 June, and is quickly followed by the third Test at Trent Bridge on 25 June with England leading the three-match series 1-0.
England Test squad: Joe Root (captain), Rehan Ahmed, Jofra Archer, Sonny Baker, Shoaib Bashir, Jacob Bethell, Harry Brook, Jordan Cox, Ben Duckett, Matthew Fisher, Emilio Gay, James Rew, Ollie Robinson, Jamie Smith (wk), Josh Tongue.
Cricket Australia is exploring new ways to strengthen its footprint in India, with chief executive Todd Greenberg suggesting the country could even host a Test match between Australia and England.
Jacob Bethell is a doubt to start for England in the first Test against Pakistan next month after withdrawing from the Hundred due to a knee injury.
Brendon McCullum says he is looking forward to working with new England Test coach as they look to achieve success in all three formats of the game.
England smashed the highest one-day international total ever recorded at Lord's before holding off a spirited India chase to secure a 27-run victory and a 2-1 series triumph over the world's top-ranked side.
