Rev Bryan Kerr’s motion, as amended, was passed 345 to 170 at the General Assembly in Edinburgh on Saturday.
The Legal Questions Committee now has two years before it has to report back, with a final poll expected in 2021.
Speaking before the vote, Rev Tom Gordon said: “I have two daughters, both of whom are married.
“I have one gay daughter in a same-sex marriage.
“When my older daughter got married she had a choice - to ask me to conduct her service as a minister or for me to walk her down the aisle as her dad.
“But when my younger daughter got married, she had no such choice.
“Give people a choice - allow their marriage to be conducted by a minister.
He added: “With God’s blessing.”
Fears were raised by others in the assembly about whether relevant safeguards would be in place to protect those who did not wish to perform the ceremonies.
Rev Mark Malcolm claimed he would be willing to “give up” his right to conduct all marriages and only do blessings, as a compromise for the “peace of the church”.
He said people who do not support same-sex marriage within the church have been described as “bigots”.
Rev Malcolm added: “Legislation isn’t going to protect, because perception is everything.”
The amended motion, as passed, outlines that the legislation would allow ministers who wish to perform same-sex marriages to do so.
It also sets out that the draft law should provide safeguards for those who do not wish to.
In June last year, the Scottish Episcopal Church took the “momentous” decision to allow same-sex couples to be married in church.
The historic move made it the first branch of the Anglican faith in the UK to allow same-sex marriage.
The General Assembly meets for a week every year in May.
It has the authority to make laws determining how the Church operates and can also act as the Kirk’s highest court.
This year’s Assembly closes on May 25.