Atkinson pulled up holding his hamstring during the fourth Test against Australia in Melbourne and has subsequently been sent home.
He becomes England's third fast bowler to leave the tour through injury after Mark Wood (knee) and Jofra Archer (side strain).
With only single Test remaining in the series, the tourists have opted against drafting in a replacement into the squad.
Atkinson played the first two Ashes Tests but struggled and was dropped for the game in Adelaide as England went 3-0 down to lose the series.
He came back into the XI at the MCG when Archer suffered his injury and took three wickets on a spicy pitch as the visitors won by four wickets.
Matthew Potts and Matthew Fisher are the other seamers with the squad who have not featured in the previous four Test matches.
Shoaib Bashir is also an option if England want another spinner to accompany Will Jacks, although that would be unlikely.
The International Cricket Council has rated the pitch used for the Boxing Day Ashes Test at the Melbourne Cricket Ground as "unsatisfactory", issuing the venue with one demerit point.
England captain Ben Stokes was happy that his team broke their Ashes drought in Australia but slammed the state of the pitch in Melbourne after the Boxing Day Test ended in two days.
England ensured the Ashes would not end in a clean sweep by securing a four-wicket win over Australia after just two days in the fourth Test at Melbourne.
England were skittled for 110 inside 30 overs on the opening day of the Boxing Day Ashes Test in Melbourne, after earlier bowling Australia out for 152 thanks to Josh Tongue's five-wicket haul.
