Health Secretary Matt Hancock said he was "very sorry" for breaching coronavirus social distancing rules after photos surfaced of him kissing a close aide.
Photos were published by the Sun appearing to show him in an embrace with his aide, Gina Coladangelo, on May 6 when Government guidance said two people from different households should not hug.
In a statement, Mr Hancock said: "I accept that I breached the social distancing guidance in these circumstances.
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"I have let people down and am very sorry.
"I remain focused on working to get the country out of this pandemic, and would be grateful for privacy for my family on this personal matter."
In the statement, the married Health Secretary confirmed he has no intention of resigning despite numerous calls for him to do so.
Labour demanded he is sacked as Health Secretary over a "charge sheet" that culminated in the incident.
The images, apparently captured from CCTV footage, were taken on May 6 at the headquarters of the Department of Health and Social Care (DHSC), the Sun said.
The minister hired Ms Coladangelo as an unpaid adviser on a six-month contract in March last year, before appointing her as a non-executive director at the department.
Mr Hancock, who met Ms Coladangelo at Oxford University, where they both worked on the student radio station, has been married to his wife Martha for 15 years and they have three children.
Ms Coladangelo is the marketing and communications director at Oliver Bonas, a British retailer founded by her husband, Oliver Tress.
Liberal Democrat leader Sir Ed Davey said on Twitter: "The reason Matt Hancock should resign is that he is a terrible Health Secretary, not because of his private life.
"From the PPE scandal, the crisis in our care service and the unbelievably poor Test and Trace system, he has utterly failed."
Meanwhile, Labour said the Government needs to answer whether the Health Secretary had broken any rules or there had been "conflicts of interest" in the appointment of his adviser.
Transport Secretary Grant Shapps said on Friday morning that reports about his Cabinet colleague were a personal matter and that Ms Coladangelo would have gone through an "incredibly rigorous process" to get the role.
Asked if the Health Secretary had been ignoring Covid-19 social distancing rules when the images were taken, Mr Shapps told LBC: "I’m quite sure that whatever the rules were at the time were followed.
"You’ll recall that there was a point at which social distancing rules were changed but, as I say, I don’t want to comment on somebody else’s private life – that is for them."
However, the Government’s road map out of lockdown said people should continue to keep their distance from anyone not in their household or support bubble until May 17.
Asked about the rules around appointing friends to Government positions, Mr Shapps told Sky News: "First of all, I think the actual issue is entirely personal for Matt Hancock.
"In terms of rules, anyone who has been appointed has to go through an incredibly rigorous process in Government, so, whatever the rules are, the rules will have to be followed.
"There are no short cuts to that, as anyone who has had anything to do with the appointments system in the Civil Service knows."
He said there are "very strict rules in place" in terms of how advisers are appointed, adding: "I think it is a bit of red herring in this case."
Responding to the reports, a Labour spokesman said: "Ministers, like everyone, are entitled to a private life.
"However, when taxpayers’ money is involved or jobs are being offered to close friends who are in a personal relationship with a minister, then that needs to be looked into.
"The Government needs to be open and transparent about whether there are any conflicts of interests or rules that have been broken."
