The SNP MSP for Glasgow Shettleston made the claim when he discussed the murder of three unarmed Scottish soldiers at the hands of the terrorist organisation during the Troubles.
A relative of one of the murdered soldiers has described Mr Mason’s conduct as an insult to his family and demanded that Ms Sturgeon says sorry for the behaviour of her colleague.
The Scottish Conservative Party has put down a motion at Holyrood also demanding an apology from the SNP.
Mr Mason made his comments when asked to support a justice campaign for three soldiers killed in 1971 while off-duty.
Teenage brothers John and Joseph McCaig, aged 17 and 18, along with 23-year-old Dougald McCaughey were serving in the Royal Highland Fusiliers when they were lured from a Belfast bar.
They were taken to a lonely spot on the outskirts of Belfast and executed. Their bodies were found by children playing the next day.
A cousin of Mr McCaughey told BBC’s Good Morning Ulster that his family had asked Scottish First Minister and SNP leader Nicola Sturgeon for a public apology.
David McCaughey also said he wanted to the SNP to take disciplinary action against Mr Mason.
Mr Mason was asked on Twitter to support a campaign to bring the killers to justice, with the message: “Will you support this campaign, or are these brave lads not Scottish enough in your opinion?”
Mr Mason responded: “Happy to support all campaigns to bring about justice. But not taking sides between Irish and British.”
After being challenged about “not taking sides between Scottish soldiers and Irish murderers”, he replied: “You say Irish murderers. Others say Irish freedom fighters. I support Scottish soldiers if they do good but not if they do bad.”
His comments caused outrage. Mr Mason later tweeted: “I have been asked to clarify my comments were general and not about any specific case. I certainly did not intend to cause any offence”.
Mr Mason has declined to comment further on the remarks.
Mr McCaughey said Mr Mason’s tweets were “deplorable” and made him “sick to the stomach”.
“To turn around and say an organisation like the Provisional IRA are freedom fighters - that means you’d need to put ISIS, the Basque separatists and any other terrorist organisation down this freedom fighter route,” he said.
“It was an insult to my family and also to other people who have lost their lives at the hands of the IRA.”
“The man should be ashamed to be Scottish,” he added.
A spokesman for the Scottish Conservatives said: “As Mr McCaughey says this morning, John Mason’s comments on Twitter were an insult to his family and to the many, many other families who lost a loved-one at the hands of the IRA. “As Mr McCaughey also says, neither Mr Mason nor Nicola Sturgeon has apologised for these offensive comments. They should do so in order to reaffirm that we all stand against the terrorists who brought so much tragedy and bloodshed to Northern Ireland.” Remarkable Voices: find out more about the people who have shaped Scotland