Mr Newton was dismissed from his post in January after allegations of gross misconduct, but reinstated last month after a successful appeal.
His secretary, suspended in October at the same time as Mr Newton, has had her suspension lifted but remains absent due to sickness.
Unison, which is representing her, has sent a formal letter of complaint to the council's only other chief officer, director of resources and housing Eric Fisher. It asks for an independent investigation of five allegations against chief executive Ken Whitehead.
The council announced on Tuesday it had decided to terminate Mr Newton's employment 'in the interests of the efficient exercise of the council's functions'.
This means he will be entitled to an immediate pension with added years.
Mr Whitehead declined to comment on the matter.