Fiona McLeod, SNP for MSP for Strathkelvin and Bearsden and a supporter of the change in the law, was asked for a meeting by constituent Rose Docherty from Woodhill, Bishopbriggs.
Mrs Docherty, a retired local authority employee and former SNP member, responded to a letter on the forthcoming referendum urging constituents to return a pre-paid postcard detailing any areas of concern about the referendum or call if they wished to discuss any other matters.
She said: "I was refused an appointment to meet with Fiona McLeod MSP to discuss this issue. I asked why and was told because she has already firm views on this issue and won't be changing her opinion.
"When I clarified that in effect my local MSP was refusing an appointment with a constituent to discuss an issue as a result of having been encouraged to get in touch, I was told that that was correct. I find it quite astonishing."
A Scotland For Marriage spokesman said: "It is astonishing that any MSP would refuse to meet with her constituents to discuss a Bill which is being considered by the Scottish Parliament."
A spokesman for Ms McLeod said: "Any complaint to the Presiding Officer is a matter for her, and cannot be commented on by an MSP until it has been resolved."
Section 8 of the Code of Conduct covers dealings with constituents and does not appear to put a duty on MSPs to agree to meetings.