The prime minister's chief aide is still facing calls to resign after allegations he broke the coronavirus lockdown restrictions.
Dominic Cummings said he believed he acted "reasonably" and within the law after driving 260 miles from London to Durham in March.
At a press conference in Number 10 Downing Street's garden, Mr Cummings said he made the trip over fears of a lack of childcare if he came too ill with Covid-19 to look after his young boy.
He also revealed concerns about his family's safety but said despite the backlash, he had "not considered" resigning.
Boris Johnson attempted to shift focus away from the row by announcing all shops across England will be able to open from June 15 if they can meet the coronavirus guidelines to protect shoppers and workers.
Opposition leaders will hold talks on Tuesday morning to discuss public health messaging amid the row.
Outlining his trip to the North East, Mr Cummings told reports how his wife fell ill on March 27, leaving him to swiftly leaving Number 10 to return home.
After a couple of hours, she felt better and Mr Cummings went back to Downing Street.
Later than evening, he discussed the situation with his wife, including the fact that many people inside Number 10 had developed coronavirus symptoms.
Mr Cummings said he was worried that if they both fell ill, there was “nobody in London we could reasonably ask to look after our child and expose themselves to Covid”.
The family drove to Durham that evening and did not stop on the way, he said. They stayed in an “isolated property” on his father’s farm, where the following day he woke up in pain and “clearly had Covid symptoms”.
Mr Cummings said that by April 11 he was still feeling “weak and exhausted” but had no coronavirus symptoms, so thought he would be able to return to work the following week – possibly part-time.
But he said that because his eyesight had been affected by the disease, his wife did not want to risk the long drive back to London, so they went on a “short drive” to Barnard Castle – around 25 miles away from where he was isolating.
“We did not visit the castle, we did not walk around the town,” he insisted, but said that he had felt a “bit sick” so they had walked about 10 to 15 metres to the riverbank where they sat for about 15 minutes until he felt better.
The family returned to London on April 13, and he went back to work the next day, Mr Cummings said.
In the hour-long press conference, he declined to apologise for his actions, but conceded that “reasonable people may well disagree about how I thought about what to do in the circumstances”.
However, he said: “I don’t regret what I did… I think what I did was actually reasonable in these circumstances. The rules made clear that if you are dealing with small children that can be exceptional circumstances.
“And I think that the situation that I was in was exceptional circumstances, and the way that I dealt with it was the least risk to everybody concerned if my wife and I had both been unable to look after our four-year-old.”
At least 15 Conservative MPs said Mr Cummings should go, while several others have spoken out against his actions.
A spokesman for the Labour Party said following the press conference: “The British people were looking for at least an apology from Dominic Cummings for breaking the lockdown. They got none.
“Millions of people have made extraordinary sacrifices during the lockdown. Families have been forced apart, sometimes in the most tragic of circumstances. They stayed at home to protect the NHS and save lives.
“And yet, the message from this Government is clear: it’s one rule for Boris Johnson’s closest adviser, another for everybody else."
Coronavirus: Everything you need to know