The leader of the Scottish Conservatives has told Downing Street that Dominic Cummings should consider his position, BBC Scotland has learned.
Jackson Carlaw made his views known as Scotland Office minister Douglas Ross quit the UK government over the row.
The Moray MP said "events over the last few days mean I can no longer serve as a member of this government".
Mr Cummings - a senior adviser to Prime Minister Boris Johnson - has defended travelling to County Durham in March.
But Mr Carlaw said the row was proving to be too much of a distraction in the effort to tackle coronavirus,
He had earlier backed the prime minister's decision not sack his aide but said he "entirely respected and understood" Mr Ross's decision to stand down.
In a statement announcing his resignation, Mr Ross said he could not justify Mr Cummings' actions to his constituents.
He said many of them have had to miss funerals and seeing sick family members during the coronavirus lockdown, and that "I cannot in good faith tell them they were all wrong and one senior advisor to the government was right".
A No 10 spokesman said: "The prime minister would like to thank Douglas Ross for his service to government and regrets his decision to stand down as Parliamentary Under Secretary of State for Scotland."
A series of senior government ministers have rallied around Mr Cummings, who insisted he acted reasonably and legally by going to stay on his parents' farm during the lockdown when he, his wife and young child were ill.
In his resignation letter, Mr Ross said he accepted that Mr Cummings had acted "in what he felt were the best interests of his family", but that "these were decisions many others felt were not available to them".
He added: "While the intentions may have been well meaning, the reaction to this news shows that Mr Cummings' interpretation of the government advice was not shared by the vast majority of people who have done as the government asked.
"I have constituents who didn't get to say goodbye to loved ones; families who could not mourn together; people who didn't visit sick relatives because they followed the guidance of the government.
"As a father myself, my instinct is to always do what is best for my son and wife. We have been fortunate not to have caught this awful virus but, if we did, we are prepared to follow the government advice and stay at home to contain this virus."
Douglas Ross backed Boris Johnson for the Conservative leadership and his approach to Brexit, and was rewarded with a job in government after the 2019 general election.
His resignation comes as a surprise. But as well as being a politician, Mr Ross is also a qualified football referee who is in the habit of calling out foul play.
As a junior minister, he doesn't have the power to discipline a special adviser to the prime minister.
But by quitting government himself - with the devastating analysis that he could not in good faith justify Mr Cummings' behaviour to constituents who stayed away from sick and dying relatives - Mr Ross has heaped fresh pressure on Boris Johnson to sack him.
His courage in taking a stand may also embolden other angry Conservatives to speak out too.
At a news conference in the garden of 10 Downing Street on Monday afternoon, Mr Cummings said he did not regret his actions in travelling 260 miles with his ill wife and child to his parents' farm in County Durham on 27 March, at the height of the lockdown.
Mr Cummings said he became ill the morning after arriving at the farm, and his four-year-old son subsequently spent a night in hospital.
After staying at the farm for a fortnight, Mr Cummings said he drove his family 30 miles to the town of Barnard Castle because he had been suffering vision problems during his illness and wanted to test his eyesight to see if he could make the journey back to London.
The prime minister said he regretted the "confusion and anger" caused by the row, but has continued to back his top adviser.
Cabinet Cabinet Office Minister Michael Gove said Mr Cummings' account of his actions was "exhaustive, detailed and verifiable" and "people will make their own mind up".
Opposition parties including Labour, the SNP and the Lib Dems accused Mr Johnson and Mr Cummings of double standards, and are to meet on Tuesday to discuss the next steps in the row.
Fellow Tory MSP Adam Tomkins said Mr Ross's resignation was "a disaster" and "shows exactly why Cummings should be sacked", adding: "I suspect others will follow where Douglas has led."
I haven't commented publicly on the situation with Dominic Cummings as I have waited to hear the full details. I welcome the statement to clarify matters, but there remains aspects of the explanation which I have trouble with. As a result I have resigned as a government Minister. pic.twitter.com/6yXLyMzItJ— Douglas Ross MP (@Douglas4Moray) May 26, 2020
First Minister Nicola Sturgeon - who has been critical of Mr Cummings - said "fair play" to Mr Ross at her daily media briefing.
She said: "I think like what I suspect is the majority of the population, he believes that Dominic Cummings' actions were not acceptable, and the handling of that and the retrospective rewriting of the rules to try and somehow justify it is not acceptable either.
"I think he's taken a principled position by resigning, and we'll see whether others decide to follow suit."
Ian Murray, Labour's Shadow Scottish Secretary, described Mr Ross as a "fair and reasonable minister who has done the decent thing and resigned from a government that is out of control".
Who is Douglas Ross?
Formerly a Moray councillor and member of the Scottish Parliament, Mr Ross became an MP in 2017 when he unseated the SNP's Westminster leader Angus Robertson.
He held the seat with a reduced majority in 2019, and entered government as a replacement for former colleague Colin Clark, who lost his seat.
The 37-year-old is also a qualified football referee, and has acted as a linesman in a number of high profile fixtures including Scottish Cup finals and Champions League matches.
However he stepped back from most footballing duties after controversy arose over him missing a vote in the House of Commons to run the line at a Barcelona match.