Richard Leonard's party has now formally adopted a clearer stance on the issue and is in favour of a public vote being held on any Brexit deal - in which it will campaign to remain.
After that policy switch, the First Minister urged her rival to consider changing his position on a second independence vote.
He challenged her on the issue after the BBC announced it would no longer provide free TV licenses for all those aged over 75.
Mr Leonard told the SNP leader: "The way we treat our elderly citizens is a mark of the kind of society we are, these are people who have contributed all of their working lives, many are still contributing today as unpaid carers, yet too many are forced to choose between heating and eating.
"First Minister, your target date to end fuel poverty was 2016, now it's 2040."
Ms Sturgeon said issues such as regulation of energy prices and pensions were all controlled by Westminster.
"Let me say to Richard Leonard the regulation of energy prices in this country is a reserved matter," she said.
"Pensions are a reserved matter, television licences are reserved matter.
"So if Richard Leonard wants this Government to have responsibility for all of these matters he will find I am the first to agree with him.
"Having reversed his position on a second EU referendum at the weekend maybe he will now see the light and reverse his position on a second independence referendum so this Parliament can take control of these matters out of the hands of the Tories."