Ms Szubanksi, who came out in 2012, asked Ms Nash: "Do you think I'm equal to you?"
"Of course I do," Ms Nash replied.
"If I was your daughter, and I'm being gay, would you think that I should have the right to be married?"
Ms Nash answered: "My view is still the traditional view of marriage. I love my children regardless of what they ever brought home for me, would make absolutely no difference at all.
"I completely respect your view and your desire to see that as equality."
"But you won't give me my rights. Thanks for nothing," Ms Szubanksi responded.
Ms Szbunaski illustrated her point by describing the families of her and Australian rock star Jimmy Barnes', another member of Monday's panel.
Senator Fiona Nash and Actor Magda Szubanski. Source: ABC Australia
"Now, Jimmy and I are family. I'm the godmother of his granddaughter.
"He's a Scottish migrant, I'm a Polish-Scottish migrant. His wife is Thai. I'm a lezzo. We are that modern family. What threat does it pose except I don't have the same rights as other people?" she asked.
Mr Barnes echoed her views, saying people had the "right to be free and to have a choice".
Ms Szubanski said discrimination against her sexuality led her to suicidal thoughts when she was a child.
"I did, when I was 11, 12, 13, and knew that I was gay and this was not going to change and I had no control of it, I knew that it would mean an absolutely terrible life," she said.
"As an LGBTQI person, you're a minority of one, within your own family. It is such a precarious feeling."
Social media users responded with an overwhelming vote of support for Ms Szubanski.
The Australian government is proposing to hold a plebiscite, a non-binding national vote, to determine whether to legalise same-sex marriage. The Labor party is yet to confirm its stance about whether there should be one, with Opposition Leader Bill Shorten in late August describing it as a "second-best option".
A plebiscite has been estimated to cost about $160 million.
