The Labour leader said the Prime Minister must publicly confirm that former Cabinet minister Mr Paterson will not be nominated for a peerage.
Ahead of an emergency Commons debate on standards at Westminster, Sir Keir said Mr Johnson needed to “clean out the filthy Augean stable he has created”.
He also called for action against disgraced MP Rob Roberts, who was readmitted to the Conservative Party despite breaking Parliament’s sexual misconduct policy.
Although he is back in the party, the Delyn MP sits as an independent in the Commons as the Conservative whip remains suspended.
A Commons debate last Monday was granted by Speaker Lindsay Hoyle following Tory attempts to block an immediate 30-day suspension for Mr Paterson over an “egregious” breach of lobbying rules.
Conservative MPs were ordered to back the creation of a Tory-led committee to look again at Mr Paterson’s case and the whole standards system.
But after a backlash over the plan the Government performed a U-turn and Mr Paterson subsequently quit as an MP, leaving what he called the “cruel world of politics”.
Sir Keir said: “Boris Johnson needs to attend this debate, answer for his mistakes, apologise to the country and take action to undo the damage he has done.
“The country is yet to hear a word of contrition over his attempts to create one rule for him and his friends and another for everyone else. He must now come to the House and say sorry.
“And he needs to go beyond just words. Today, the Prime Minister must begin to clean out the filthy Augean stable he has created.”
The reference to cleaning the Augean stable will be familiar to the classics-loving Prime Minister, as it was one of the labours of Hercules.
Sir Keir, who will lead Labour’s response in the Commons debate, said Mr Johnson should confirm he will not nominate Mr Paterson “or any other MPs who have been handed suspensions from Parliament” for a peerage.
Downing Street sources have privately indicated there is no prospect of a seat in the Lords for Mr Paterson after speculation in Westminster that he would receive the honour.
In on-the-record comments on Friday a No 10 spokesman refused to rule out the possibility, saying only that there was a “formal process”.
He also urged Mr Johnson to work together on plans to throw Mr Roberts out of the Commons – a loophole meant the Delyn MP’s six-week suspension could not trigger the recall process which leads to a by-election.
Mr Roberts voted with Tory MPs for the plan to spare Mr Paterson an immediate suspension.
Sir Keir said: “It is disgraceful that Mr Roberts has been welcomed back as both a Member of Parliament and the Conservative Party despite having been found to have sexually harassed a junior member of staff.
“That he was able to aid and abet the Prime Minister in his attempts to corrupt British politics last week should be a source of shame to the Tories.
“The Prime Minister was prepared to rip up the system to save one of his disgraced MPs – why will he not take action to protect others from this one?”
Sir Keir also demanded a “full, transparent investigation” into how Randox – one of the firms that paid Mr Paterson – came to win Covid-19 testing contracts.
He said it was “vital the public has confidence that Owen Paterson’s paid advocacy did not influence these decisions”.
The Parliamentary Commissioner for Standards’ investigation into Mr Paterson’s activities covered from October 2016 and February 2020, before the pandemic struck.
Sir Keir said if Mr Johnson did not agree to his demands it would “send a message that the Prime Minister is either too arrogant or too cowardly to take responsibility for his own behaviour”.
“It will show that he believes there should be one rule for him and another for everyone else, and that the Tories are concerned with their own interests, not those of the country.”
The Liberal Democrats, who secured the emergency debate, have called for an independent statutory public inquiry into sleaze and corruption allegations.
The inquiry, which would have the power to summon witnesses and take evidence under oath, would examine not only the Paterson row but also the awarding of coronavirus contracts, whether Mr Johnson’s holidays were properly declared and the refurbishment of the Downing Street flat.
The party also said that any MPs being investigated by the Parliamentary Commissioner for Standards should not be able to vote or propose amendments to motions related to disciplinary issues.
Lib Dem chief whip Wendy Chamberlain said it was “the equivalent of defendants in a court case also taking part in the jury”.
She added: “Time and again Government ministers have refused to properly investigate allegations of sleaze, failed to declare relevant meetings and donations and tried to rig the system to cover their own backs.
“We need an independent public inquiry, with the powers and resources to get to the bottom of this Conservative sleaze scandal.”