Her remarks came in response to a letter read out by Scottish Conservative MSP Graham Simpson at today’s First Minister’s Questions.
The letter from an East Kilbride policeman questioned why officers received a smaller pay rise than teachers or NHS staff.
He asked if police were regarded as an “easy target” because they are barred from taking industrial action and questioned the value the Scottish Government placed on the police service.
This year’s pay offer for police was 6.5 per cent compared with 13.5 per cent for teachers and nine per cent for NHS workers.
But the First Minister hit back at the suggestion, and said: “Parts of those comments were quite disgraceful actually. I value all public sector workers and I thank them for the work that they do.
“On the police pay award - that is the best award for police officers anywhere in the UK and the Scottish Police Federation said it was the best award in 20 years.”
She added: “If the member thinks 6.5 per cent is not good enough... then I wonder what he makes of the two per cent that has been awarded to police officers in England by his Tory colleagues in the Westminster government, I think described by the head of the Met as a punch in the nose for police officers?”
Later, Mr Simpson said the First Minister should apologise for her remarks in Holyrood.
He said: “Nicola Sturgeon should be taking the concerns of a serving police officer extremely seriously. Instead, in front of Scotland’s parliament, she branded him a disgrace.
“It’s no wonder officers are losing faith in this SNP government with that kind of attitude. These brave and hardworking police officers should not be discriminated against because they don’t threaten to go on strike.
“The First Minister needs to apologise for being so dismissive of his concerns.”
In the letter, the anonymous officer said: “Why has such a massive importance been placed on the pay rise for school teachers when other public service staff are not being awarded nearly as much as a pay rise?
“Does the Scottish Government place the value of police officers as only half that of school teachers? Is it that the Government know that because police officers cannot strike or take any real industrial action that they are an easy target?”
A spokesman for the First Minister later said that she would not apologise as her "disgraceful" comment was not directed at the police officer but at Graham Simpson.
During First Minister's Questions, Mr Simpson was booed by SNP backbenchers as he read out the letter, and asked what Nicola Sturgeon would say to the officer and “thousands of others” like him.
Ms Sturgeon went on to say: “In terms of our NHS workers many get higher pay in Scotland than they do in England because of the value we attach to the work that they do.
“In terms of teachers they have now been offered an exceptionally good pay deal and I think that is recognition of the good work that they do and I hope that is now accepted.
“But I value all public sector workers and I think if you look at any group of public sector workers you will find that the value attached to them by the Scottish Government is much much greater than the value attached to their counterparts in England by the Tory government at Westminster.”