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.
Jonathan Trott, who has been linked with the England Test coaching job, will join the Ireland staff as a consultant ahead of their ODI series against Afghanistan next month.
India made a winning start to the one-day international series with a comfortable six-wicket victory over England at Edgbaston on Tuesday, led by captain Shubman Gill and all-rounder Axar Patel.
Brendon McCullum admits his results as head coach of England's Test side were not good enough and it was probably for the best that he was relieved of his duties in the longest format.
Brendon McCullum has been dismissed as England's Test head coach, bringing the curtain down on the Bazball era just weeks after Ben Stokes retired from international cricket.
