Rick Santorum said on Sunday he would continue his campaign to reverse the supreme court decision that legalised same-sex marriage, because the US “is still the moral leader of the world”.

On 26 June, the court voted 5-4 in favor of the plaintiff in the case Obergefell v Hodges, who sought to extend rights conferred by a same-sex marriage in one state to another which did not allow it.

Since then, conservative opposition has included calls for a constitutional amendment on marriage, advocacy for laws to ensure religious liberty, and official approval of county clerks who refuse to issue licenses for same-sex marriages.

Appearing on CBS, Santorum was asked about a previous statement that the ruling “disrupts the foundation of the world”.

He said he was concerned with three things: the role of the court in society – a rallying call for 2016 candidates including Mike Huckabee and Bobby Jindal – an “assault on religious liberty” and “a further erosion of this founding building block of society, which is the nuclear family”.

“Over the last 40 years we’ve seen a degradation of the nuclear family, no doubt about it,” said the former Pennsylvania senator who is running for the White House for a second time, having finished second in the primaries in 2012. “But this further put the nail in the coffin.

“We now disconnect the nuclear family from the idea that it is there for the purpose of having and raising children … it’s [now] about adults.

“The United States is still the moral leader of the world. We’ve now disconnected marriages from having children and I think that has profound consequences not just for the United States but for the world.”

Asked if same-sex couples could not raise children to be good citizens, as straight couples could, Santorum said: “I think what we have to do as a society is orient ourselves towards what’s best.

“What we know is best from thousands of years of human history is for children to be raised with mothers and fathers, preferably but not always their biological mothers and fathers. Adoptive homes are great and wonderful places too.”

He added: “We have laws that say, ‘Fathers, really you don’t have to raise your children; mothers, you know, we’re going to provide all sorts of things that make fathers less necessary.’

“… We have now said marriage is not about having children, so people are not getting married. That’s not a good situation to maximise the potential for each and every one of our children. And that’s what I’m really talking about here.”

Santorum, a committed Christian and author of the 2005 book It Takes a Family: Conservatism and the Common Good, also said adultery and divorce were social problems that needed to be addressed.

Supreme court plaintiff Jim Obergefell receives an emotional phone call from President Barack Obama. Guardian

The argument that marriage is fundamentally about establishing a stable relationship in which to raise children was raised during the hearing of Obergefell v Hodges by lawyer John Bursch, appearing on behalf of the states trying to preserve bans on same-sex marriage.

After a debate over whether it was fair to ask couples if they wanted children before allowing them to wed, justice Ruth Bader Ginsburg, 82, who would be among the majority in the eventual ruling, prompted laughter when she said: “Suppose a couple, 70-year-old couple, comes in and they want to get married?

“You don’t have to ask them any questions. You know they are not going to have any children.”