Christoper McEleny said he was the target of malicious communications on Twitter on a daily basis, often linked to his religion.
The leader of the Nationalist group on Inverclyde Council claimed that he already chose to ignore the vast majority of the abusive messages he received.
But he found one communication – sent as a direct message on social media – so offensive that he felt he had no option but to report the matter to the police.
Mr McEleny, who has stood twice for the SNP depute leadership, reported the matter on 16 March via Police Scotland’s online portal before being asked to attend Greenock police station the following day.
The councillor told The Scotsman he was subsequently informed this week the incident would be recorded as a crime but any follow-up action was unlikely to be taken.
He decided to share his frustration with the process as he fears that religiously aggravated abuse is becoming normalised on social media.
“This leaves serious questions to be answered,” he said. “If Police Scotland views anti-Catholic online hate crime as too low priority to resource investigation, does this mean people are free to openly abuse and discriminate against the Catholic community in Scotland online without fear of prosecution?
“The rhetoric is that tackling sectarianism and online hate crime is a priority. In reality, when it’s online anti-Catholic hate crime, we are told it is too low a priority. It begs the question, is policing in Scotland institutionally sectarian?”
A police spokeswoman confirmed to The Scotsman that a complaint had been received and that enquiries were on-going.
Mr McEleny was previously offered support by Police Scotland in 2016 after he received a barrage of abuse on Facebook.
Several Scottish politicians have spoken out in recent weeks against the abuse they receive online.
Tory MSP Annie Wells said she had contacted Police Scotland over a number of threats made against her.
The list member for Glasgow claimed that in the three years since she was elected as a list MSP for the Glasgow region, she has been forced to mute more than 1,100 abusive Twitter accounts due to the violent and foul-mouthed nature of the posts.