The US transportation secretary, Pete Buttigieg, has delivered an emotional appeal for Republicans to support a law protecting same-sex marriage as it heads for the Senate.

Democrats who control Congress aim to protect same-sex marriage amid uncertainty over which privacy based rights the conservative-dominated supreme court might target next, having overturned the right to abortion last month.

Appearing on CNN’s State of the Union, Buttigieg – the first openly gay cabinet secretary – was asked about dismissive remarks about the Respect for Marriage Act, which also protects interracial marriage, made by Marco Rubio, a Florida senator.

“If he’s got time to fight against Disney, I don’t know why he wouldn’t have time to help safeguard marriages like mine,” Buttigieg said.

That was a reference to Republican attacks on Disney over its opposition to legislation clamping down on teaching about LGBTQ+ issues in schools.

“This is really, really important to a lot of people,” Buttigieg said of the move to protect same-sex marriage, which the supreme court declared legal in 2015.

“It’s certainly important to me. I started my day, as I try to do on weekends, to get Chasten [his husband] a little bit of a break and do breakfast with both of our twins.

“And listen, this no small thing as every parent of small kids knows. It was one of those days where the tray table wasn’t quite fitting into the highchair and I’m trying to make sure that they’re busy enough with their little cereal puffs to give me enough time to chop up the banana and get the formula ready.

“And … that half-hour of my morning had me thinking about how much I depend on and count on my spouse every day. And our marriage deserves to be treated equally.”

This week, 157 House Republicans voted no on the Respect for Marriage Act. Democrats in the Senate hope at least 10 Republicans will back the bill, thereby beating the filibuster under which the minority can block most legislation.

“I don’t know why this will be hard for a senator or a congressman,” Buttigieg said.

“I don’t understand how such a majority of House Republicans voted no on our marriage as recently as Tuesday, hours after I was talking about transportation policy, having what I thought were perfectly normal conversations with many of them on that subject, only for them to go around the corner and say that my marriage doesn’t deserve to continue.

“If they don’t want to spend a lot of time on this, they can vote yes and move on. And that would be really reassuring for a lot of families around America, including mine.”

Liz Cheney of Wyoming, a prominent conservative nonetheless outside the pro-Trump mainstream of the Republican party, told CNN: “Freedom means freedom for everybody. And I’ve said that that my initial opposition 10 years ago to same-sex marriage was wrong.”

Cheney opposed same-sex marriage despite her own sister being gay.

“I love my sister very much and her family very much,” she said. “And I believe that that given the decision we saw from the supreme court and the suggestion that the additional rulings could be at risk … and Justice Thomas’s opinion about that means that we’ve got to step up and make sure that we’re providing protections.”

Clarence Thomas, an arch-conservative, suggested rights including same-sex marriage and access to contraception could be reconsidered. The justice, who is Black, did not mention interracial marriage. His wife, the far-right activist Ginni Thomas, is white.

Cheney said: “Ensuring that we’ve we’ve provided that kind of protection for same-sex marriage is very important. And so I would urge my Republican colleagues in the Senate to do the same.”