Changes to the appointment process for Scotland’s police watchdog need to be considered in the “fullness of time”, Nicola Sturgeon has said.

Despite other public watchdogs having “preferable” appointment processes, the First Minister pointed out a new chair has taken over at the Scottish Police Authority (SPA).

Scottish Conservative leader Ruth Davidson pressed the SNP leader to change the law to bring the SPA into line with other organisations, where appointments are made by the Scottish Parliament and not by ministers.

She challenged the First Minister on the issue 24 hours after the resignation of Phil Gormley as chief constable of Police Scotland.

Speaking at First Minister’s Questions, Ms Sturgeon said this would allow policing to “move forward”.

Mr Gormley had denied several allegations of gross misconduct, but his decision to step down means the Police Investigations and Review Commissioner (Pirc) will no longer continue its investigation.

Ms Davidson said it was an “obvious flaw” that the “head of the Scottish Police Authority is supposed to be independent of government, yet it is the Justice Secretary that appoints them”.

Ms Davidson questioned whether that was “true independence”, adding: “As this affair has shown us that same Justice Secretary can pull the head of the Scottish Police Authority into a room and make him change his mind.”

Andrew Flanagan, who preceded Susan Deacon as SPA chair, reversed a decision to allow Mr Gormley to return to work after a period of special leave after conversations with Michael Matheson – claiming that the Justice Secretary had told him this was a “bad decision”.

Ms Davidson said: “If the First Minister is serious about strengthening the structure and oversight of the single police force then having its chair appointed by Parliament and not at the grace of ministers … is a good place to start.”

Ms Sturgeon told her primary legislation would be needed to do that, and the Tory leader responded: “Guess what First Minister, this is a Parliament – changing the law is what we do.”

Ms Deacon took over as SPA chair in December and Ms Sturgeon said she was “doing an excellent job”.

The First Minister said: “Right now we have a new chair of the Scottish Police Authority in place, she is at the start of her term in office, I think she is doing an excellent job and I think we should get behind her in that.

“I think we should consider in the fullness of time before we come to appoint a new chair, whether there is changes necessary.”

Meanwhile Scottish Labour leader Richard Leonard called on Ms Sturgeon to consider the findings of a review of policing led by former Labour MSP and former police officer Graeme Pearson.

The 2015 review made 10 recommendations including improved parliamentary oversight and staffing support.

“Week after week the First Minister stands up in this chamber and demands solutions from opposition parties, solutions to problems which her government has created in the first place,” Mr Leonard said.

“Labour offered 10 but her Justice Secretary ignored them.

“Since then two chief constables have gone, morale amongst rank-and-file officers has sunk, public confidence has declined, and all the time the First Minister refuses to take responsibility.”

He added: “Will she look again at the recommendations of the Pearson review and will she find a new Justice Secretary to deliver them?”

Ms Sturgeon said: “There has been a governance review under way, that will report shortly and all of us right across the Parliament can consider any proposals and suggestions that come forward as part of that.”