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.
Ben Stokes' could be under pressure to be retained as England Test captain after it was revealed that he and teammate Gus Atkinson breached team protocols following the victory against New Zealand at Lord's.
England captain Ben Stokes has said that the "extreme conditions" at Lord's in the first Test against New Zealand will not help with the future of Test cricket.
England launched their post-Ashes rebuild in emphatic fashion with a 115-run victory over New Zealand in the first Test at Lord's, wrapping up the win before lunch on the fourth day after a brutal, seam-dominated contest.
England moved closer to victory over New Zealand after a rain-hit third day of the first Test at Lord’s, which saw just 10 overs of play possible.
