The Metropolitan Community Church (MCC) made the call before the Parliament’s Public Petitions Committee yesterday.
The church already conducts blessings for gay partners and believes they should be allowed to be wed by a religious minister. A Bill being considered at Westminster would permit civil registrars to record same-sex relationships and grant the same legal rights enjoyed by married couples.
The MCC wants the laws to go further and accused the Executive of trying to "squeeze" churches out.
But the committee made no clear statement for or against the demand and instead passed the matter on to the Justice 1 Committee, which is considering Westminster’s Civil Partnership Bill. MSPs are keen to avoid another clash with church groups similar to the bitter battle over the scrapping of Section 28, which banned education on homosexuality in schools.
The changes are being proposed for England and Wales and the Scottish Executive, which supports the moves, wants Westminster to legislate for Scotland. The MCC is a worldwide Christian denomination founded in 1968 by gay, lesbian, bisexual and transgender people of faith. The Edinburgh branch was founded in 1995 and its 30 congregation members worship at the Augustine United Church on George IV Bridge.
Stephen Harte, community ministry co-ordinator, said: "We conduct same-sex blessings all the time. They are for couples who want to ground their relationships within their religion. We are speaking up for people of faith who are in same-sex relationships.
"But the Executive is trying to squeeze the church out of this matter. It is an issue of religious freedom. If you hold a wedding in a church, it’s still recognising civil law in the same way as a wedding held in a register office. We want people to be able to make the choice which matters to them.
"Some churches do not support same-sex relationships but they should give their support to the church remaining an important part of Scottish life. We believe the Executive are trying to take that away and this could damage the church in general."
Mr Harte added that the proposed Bill would undermine the human rights of gay people of faith and warned that they would have to take their battle to the courts if the parliament refused to enforce those rights.
"The blessing ceremonies we carry out are different for each couple as it depends on what they want," said Mr Harte.
"We will continue to campaign for a change in the law to recognise the rights of same-sex couples to be married in a church and have their faith acknowledged."
Mr Harte said that the MCC had received letters of support from Church of Scotland and Methodist ministers in the Capital, who want to carry out same-sex marriages in their churches.
Jackie Baillie and Susan Deacon, Labour members of the Petitions Committee, urged the MCC to lobby MPs and the House of Lords, since the legislation was going through Westminster.
Labour committee convener Michael McMahon said: "If we write to the Justice 1 Committee and give them our official report as we always do and ask that they take on board the points raised this morning, those are the questions that would be put to the minister."