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.
Jordan Cox and Sonny Baker are set to make their Test debuts for England against New Zealand after the hosts announced their XI ahead of the second game of the series, starting at the Oval on Wednesday.
England have added seamer Henry Crocombe to their squad ahead of the second Test against New Zealand after Ollie Robinson experienced knee soreness in training and is set to have a scan.
Former England captain Michael Vaughan believes it's an overreaction to remove Ben Stokes as Test captain for breaking curfew.
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.
