The Labour leader said the Prime Minister's "unwillingness or inability" to apologise for or explain the reason behind Mr Cummings's journey to Durham from London made the Government look "untrustworthy" and "unprincipled".
Writing in the Daily Mirror, he said the Government had “undermined the very public health advice that is necessary to keep us all safe, just to keep one powerful aide in his job".
It comes amid a growing revolt within the Conservative Party over the issue, with dozens of backbench Tories criticising his actions, and at least 38 calling for him to quit or be sacked. Senior minister Penny Mordaunt also admitted there were “inconsistencies” in Mr Cummings’ account – saying “there is no doubt he took risks”.
However, Mr Johnson has continued to stand by his aide and insisted it was time to “move on” when he faced intense questioning over the issue in an appearance before the Commons Liaison Committee of senior MPs on Wednesday.
Sir Keir said: "Let’s begin with something Boris Johnson can’t bring himself to say but the rest of us know: Dominic Cummings broke the rules. It is as simple as that.
"His actions were an insult to the millions of people across Britain who have sacrificed so much during this pandemic."
The Labour leader said lots of people in the UK had obeyed the lockdown rules - despite how hard it has been - because "they understand the importance of our shared common goal" to stay home and protect lives.
He continued: "That’s why, when the revelations about Dominic Cummings’ behaviour broke over the weekend, people were angry. And during the endless hours of press conferences over the Bank Holiday weekend, people were looking for an apology, a resignation or a plausible explanation from Boris Johnson and Dominic Cummings about what happened -None was forthcoming.
"Instead, we got a very simple message: it is one rule for them, another rule for everybody else.
"I share people’s frustrations, not just because of the actions of one adviser, but because this has become a national distraction at a moment of national crisis."
He added: "This was the week when we should have been talking about how we ease the lockdown safely. How we restart our economy, support businesses, get more children back to school.
"Instead, the Prime Minister’s unwillingness or inability to do the right thing has left the Government looking untrustworthy, unprincipled.
"And worst of all they have undermined the very public health advice that is necessary to keep us all safe, just to keep one powerful aide in his job."
Sir Keir said, at times like this, the nation needs a good Government it can trust which is focused on saving lives.
He added: “So, I say to the Prime Minister: We cannot go on like this. We cannot lose another week to this farce. We need to get back on track.”
Labour has so far held off on calling for Mr Cummings resign. Sir Keir also declined to join a Tuesday morning conference call involving all the other party leaders to agree a co-ordinated response.
But, after the news emerged, Labour wrote to the head of the civil service, Sir Mark Sedwill, calling for an urgent inquiry.
Shadow foreign and commonwealth affairs secretary Lisa Nandy also said on Wednesday that the Prime Minister must account for why the trip made by his chief adviser was “so unique” that the rules on lockdown had to be broken.
She said that if Mr Johnson is unable to do so, he must take action to restore public confidence.