Ruth Davidson said people were angry at the SNP for putting "its own narrow political interests before those of the wider country".
But the SNP said Ms Davidson should be apologising for the "mess" her party has created with Brexit.
On Friday, Nicola Sturgeon launched a "new conversation" on independence.
In a speech in Stirling, the first minister urged Scotland to "control its own destiny".
She said her party's new "listening exercise" on the issue would seek the views of voters in Scotland using a survey and meetings across the country.
Ms Davidson, in a speech on Saturday to the Scottish Conservative party's convention in Perth, said: "Nicola Sturgeon said yesterday that she wants to listen to people who voted No in the hope of meeting their concerns. Well let me tell her how many of us feel.
"We do not look back at the referendum on independence with much in the way of nostalgia. It wasn't 'civic' or 'joyous' as the SNP have claimed.
"Instead, we remember the division it caused, we remember the pain felt by many, and we remember the fear that our country was about to be split up.
She went on: "So if the first minister wants to know the feelings of thousands of Scots across the country - I can give it to her right now.
"It is anger at an SNP government which has both broken its word and decided to put its own narrow political interests before those of the wider country.
"And it is frustration that we have a nationalist administration which is acting like a drag on Scotland's progress.
"Instead of a proper government using new and existing powers to drive the country forward, we instead have a nationalist administration dragging Scotland backwards - allowing the lead weight of separatism to threaten our ambitions and hopes."
The SNP described Ms Davidson's speech as "arrogant hypocrisy".
A spokesman said: "It was Ruth Davidson who promised that 'No means we stay' in the EU and she should be apologising for the mess her party has now created with Brexit.
"People in Scotland remember very well the promises that the Tory-led No campaign made in the run up to the 2014 referendum - on shipbuilding, on the EU, on devolution and more - and they have watched as one-by-one these promises and pledges have been broken.
"Scotland faces being dragged out of the EU against our will by an increasingly right-wing, reactionary Tory party."
He added: "It is only right that the Scottish government examines all options to protect Scotland's interests and maintain our place in the EU - including the possibility of a second independence referendum."
Responding to Ms Davidson's comments, Scottish Labour said the Tories were "more obsessed with independence than the SNP".
A spokesman added: "Ahead of the Scottish Parliament returning this week Kezia Dugdale outlined plans to use the new powers of the Scottish Parliament to invest in education, stop the cuts to our NHS, create more jobs and ban fracking whilst the other party leaders are still banging on about independence.
"Labour are the only party talking about Scotland's future, rather than the arguments of the past."
Meanwhile on Saturday, pro-Europe rallies were held in cities across the UK.
More than 100 protesters gathered at the Scottish Parliament in Edinburgh to call for stronger ties with the European Union following the Brexit vote.
The organisers said all the main political parties except the Conservatives sent representatives.

Sirigu, who spent last season on loan in Spain with Sevilla and Osasuna, moves on a free transfer after ending his PSG contract by mutual consent.
England international Joe Hart was Torino's first-choice keeper last season, but has returned to Manchester City after completing a loan spell.
"The club and the president made me feel important and I thank them for that," said Sirigu, 30.
Sirigu, capped 17 times by Italy, joined PSG from Palermo in 2011, but lost his place to Kevin Trapp at the start of the 2015-16 season.

South Yorkshire Police officers were called to High Melton College at about 14:45 GMT on Tuesday where workmen had found what appeared to be a body.
Dark and cold conditions prevented officers from accessing the body so the scene was guarded overnight.
Detectives are working with Yorkshire Water, underwater search teams and crime scene officers to determine the circumstances surrounding the incident.
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High Melton is a campus of Doncaster College.
The Scottish Conservative leader has accused the Scottish government of allowing "the lead weight of separatism" to drag Scotland backwards.
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Torino have signed Italy goalkeeper Salvatore Sirigu from Paris St-Germain.
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A body has been found in a sewage tank at a Doncaster college.