Nicola Sturgeon’s party has made clear it will use an SNP victory in the Holyrood elections to push for another ballot on the future of the UK.
But with a rift in the party having been revealed, after former first minister Alex Salmond accused senior SNP figures of being part of a conspiracy against him, Sir Keir said his rivals were “fighting amongst themselves, rather than fighting for the Scottish people”.
Ms Sturgeon spent a day being questioned by MSPs on a committee probing her government’s botched handling of harassment complaints made against her predecessor, denying strongly any suggestion she was out to “get” him.
The Labour leader, however, insisted that after the coronavirus crisis, the “last thing” the country needed was “more division”.
He spoke out at an event on Thursday to launch Labour’s campaign for the May elections, which include votes for the Scottish Parliament and Welsh Assembly, as well as regional and mayoral contests in England.
Sir Keir insisted: “After everything we have been through, the last thing we need now is more division.
“Yet in Scotland the SNP are fighting amongst themselves, rather than fighting for the Scottish people.”
Sir Keir hit out the SNP, saying their 14 years in power “has seen child poverty rise and educational standards fall”.
He also said: “If you can believe it, Scotland now has the lowest life expectancy in western Europe and the highest number of drug deaths. That’s a record of shame.
“And what’s the SNP’s priority? Another divisive referendum.”
With Scottish Labour just having elected Anas Sarwar as its new leader, Sir Keir said they would “focus on what unites Scotland, not what divides us”.
That, he added, would include “a Covid recovery plan to protect our NHS” and a “a catch-up programme to tackle the SNP’s appalling failure on education”.
The speech comes almost a year after he succeeded Jeremy Corbyn as Labour leader, with Sir Keir saying since then Labour had “changed” and was “reconnecting with the British people in every region and nation of the United Kingdom”.
He insisted: “This is a different Labour Party, under new leadership and we’re making a different offer to the British people.”
Mr Sarwar, meanwhile, told Scots that “the Labour Party I lead will always be on your side”.
He insisted he was “not naïve about the scale of the challenge” facing his party, as he pledged to “work day and night to give the people of Scotland the Labour Party they deserve”.
Mr Sarwar also stressed that Scotland could not just “go back to the old arguments” after the coronavirus crisis is over.
SNP depute leader Keith Brown responded: “A vote for Labour is a vote to let Boris Johnson decide Scotland’s future.
“They’ve made it clear that anyone who supports the right of the Scottish people to choose our own future is not welcome in the party.”
“By threatening to block Scotland’s democratic right to hold an independence referendum, Labour are demonstrating the same old arrogant and ignorant approach that has alienated their traditional supporters and led to voters abandoning them in droves.”