Avon and Somerset Chief Constable Nick Gargan was suspended in May 2014 amid allegations of inappropriate conduct and, later, data protection breaches.
PCC Sue Mountstevens said he had "abused his position" by forwarding emails and "storing intimate images" on his work phone.
The news was said to come as a "huge disappointment" to Mr Gargan.
Ms Mountstevens is the first PCC to start the process of using new powers to call upon a chief constable to quit.
Turbulent times for police chief
Mr Gargan, 48, was appointed by Ms Mountstevens in January 2013, after his predecessor Colin Port quit when she asked him to reapply for his job. Mr Gargan remains on full pay but is suspended from duty, she said.
Last month, an independent QC-led panel cleared Mr Gargan of gross misconduct but found him guilty of eight misconduct charges.
The findings published on the force website largely relate to "inappropriate disclosure of information" by forwarding emails and "inappropriate use" of a police-issued phone by storing "intimate" images and text messages.
It recommended a sanction of eight final written warnings - the severest possible - but Ms Mountstevens has been petitioned by retired officers and Mr Gargan's three predecessors as chief constable, who said he should go for the good of the force.
In a statement on Wednesday, she said: "Chief Constable Nick Gargan has let down the colleagues he led and the communities he was there to protect.
"He abused his position by forwarding confidential emails, interfering with a proper recruitment process and sending, receiving and storing intimate images on his police issue phone."
She said he should have "led by example" but had instead "shown flawed judgement" and been found guilty of eight counts of misconduct "including two of discreditable conduct".
"From what I have seen and heard, he has lost the confidence of local people, police officers and staff," she said.
She said she had now "initiated the process to require him to resign".
"I think his position has become untenable," she said, owing to concern within the force "at all levels".
Her decision will be referred to Sir Thomas Winsor, Her Majesty's Chief Inspector of Constabulary.
His views and those of the PCC will then be considered by the local crime panel in a process expected to take another two months. Ultimately, Ms Mountstevens will have the final word.
On Mr Gargan's behalf, the Chief Police Officers Staff Association said he had "complied with every requirement of the protracted misconduct investigation and subsequent hearing" and had apologised for misconduct.
"He placed his fate in the hands of an expert professional panel, which made a clear recommendation that he should go back to work," the statement said.
"The overwhelming majority of the rumours surrounding the chief constable have turned out not to be true; indeed none of the initial allegations that prompted his suspension led to a misconduct finding, let alone a finding of gross misconduct or criminal behaviour."
It added that he would now wait to see under "what grounds" the call for his resignation is made.
A police chief guilty of misconduct has been called upon to resign by the police and crime commissioner.