Boris Johnson is standing by a response at PMQs to the Labour leader, after he was accused of misleading MPs in a row over care homes.
The row centred on official advice at the start of the coronavirus pandemic.
Sir Keir said the government had been “too slow to protect people”, and that up until 12 March, care homes were being told it was “very unlikely” anyone would become infected.
The prime minister said “it wasn’t true the advice said that”.
Sir Keir accused the PM of misleading MPs and asked him to return to the Commons to correct the record.
But Downing Street said the Labour leader had “inaccurately and selectively quoted from the Public Health England guidance and that was what the prime minister was referring to”.
The guidance at the centre of the row was issued on 25 February and withdrawn on 13 March, a time when the virus was not thought to be spreading in the community.
It said: “This guidance is intended for the current position in the UK where there is currently no transmission of COVID-19 in the community.
“It is therefore very unlikely that anyone receiving care in a care home or the community will become infected.”
The prime minister’s spokesman accused Sir Keir of “hiding” the first paragraph of the advice, which sets it in context of the government’s thinking at the time, in his complaint to the PM.
In his letter to Mr Johnson after the Prime Minister’s Questions exchange, the Labour leader said: “At this time of national crisis it is more important than ever that government ministers are accurate in the information they give.”
He added that: “I expect you to come to the House of Commons at the earliest opportunity to correct the record.”
It came as the prime minister announced an extra £600m to fight coronavirus infections at care homes in England.
Mr Johnson said the government had brought in the lockdown in care homes ahead of the general lockdown but that there was “unquestionably an appalling epidemic” in that setting.
He added that the number of deaths in care homes had been “too high”, but that “the number of outbreaks is down and the number of fatalities well down”.
But Sir Keir said the government had been “too slow to protect people in care homes”, after figures from the Office for National Statistics (ONS) showed at least 40% of Covid-19 deaths in England and Wales occurred in care homes.
And he asked Mr Johnson to account for 10,000 “unexplained” excess deaths in care homes in April, recorded by the ONS.
He said: “The ONS records the average number of deaths in care homes each month. The last five years the average for April has been just over 8,000.
“This year the number of deaths in care homes for April was a staggering 26,000 – that’s three times the average – 18,000 additional deaths this April.
“Using the government’s figures only 8,000 are recorded as Covid deaths, that leaves 10,000 additional and unexplained care home deaths this April.”
Mr Johnson said there “is much more to do but we are making progress” on reducing the pandemic in care homes.
Sir Keir quoted a cardiologist who had told the Daily Telegraph newspaper that hospitals had “actively seeded” the virus into the “most vulnerable” population by discharging “known, suspected and unknown cases into care homes”.
Mr Johnson said: “The number of discharges from hospitals into care homes went down in March and April and we had a system of testing people going into care homes and that testing is now being ramped up.”