The UK Labor leader made his position clear in an interview with BBC Radio Newcastle.
Scottish party leader Anas Sarwar called for a “period of calm where we focus on rebuilding our country” instead of a return to the “old arguments”.
The SNP said Sir Keir’s comments proved “beyond doubt” that independence offered the only route back into the EU.
In radio phone in, Sir Keir told a caller: “We have exited the EU and we’re not going back, and let me be very clear in the north east about that – there’s no case for rejoining.
“What I want to see now is not just Brexit done in the sense that we’re technically out of the EU, I want to make it work.
“I want to make sure we take advantage of the opportunities and that we have a clear plan for Brexit.”
SNP Westminster leader Ian Blackford insisted it was “confirmation that Labor has joined the Tories on Brexit”.
He said: “It’s now beyond doubt that independence is the only way for Scotland to regain our place in Europe. There is no way back under Westminster control.
“Brexit was imposed against Scotland’s will and it has cost us billions of pounds in long-term damage, hammering our economy, compounding the Tory cost-of-living crisis, and stripping away people’s rights.
“In a shoddy attempt to win back votes in England, Starmer’s Labor Party has completely abandoned Scotland.
“The confirmation that Labor has joined the Tories on Brexit makes it all the more essential that people in Scotland have the choice to determine our own future in an independence referendum, so we can build a fairer, greener and European future as an independent country.
“The SNP Government won the Scottish election on a cast-iron mandate to hold an independence referendum, with a record share of the vote.
“There is an overwhelming majority for an independence referendum in the Scottish Parliament and that democratic mandate must be respected.”
Challenged to reject Sir Keir’s comments by the SNP, Mr Sarwar said: “I wish we had never left the European Union and I hope we can build as close a relationship as possible going forward.
“But we have all gone through the collective trauma of Covid over the last two years.
“Our country pulled together to help us get through, but the SNP and the Tories now want us to come through that and go back to fighting with each other again.
“Independence does not mitigate the consequences of Brexit, it multiplies them.
“After all the anger and division of recent years, layered with the trauma of the pandemic, a period of calm where we focus on rebuilding our country for the future should be our focus.
“I can only credibly say let’s not come through Covid and go back to the old arguments if I include, not just the yes versus no question, but also the leave versus remain question.
“We are in the middle of a cost-of-living crisis and over 650,000 of our fellow citizens are on NHS waiting lists. Instead of the past, our focus must be on building the future.”
George Holan is chief editor at Plainsmen Post and has articles published in many notable publications in the last decade.