The Scottish Conservative leader said such a move would “destabilize the UK Government when we need to be united in the face of Russian aggression”.
Critics accused him of shamefully hiding “his lack of principle” behind the war.
Mr Ross previously called for the Prime Minister to quit over the Downing Street parties scandal, but U-turned following Russia’s invasion of Ukraine, stressing the need for unity.
However, in a sign of internal party divisions on the issue, former Scottish Tory leader Ruth Davidson, who now sits in the House of Lords, insisted Mr Johnson “should go”.
She tweeted: “Met [Police] confirms what we already knew: the PM introduced liberty-curtailing rules for public health reasons.
“This caused huge hardship for those separated from ill or dying loved ones.
“He then broke the rules he imposed on the country and lost the moral authority to lead. He should go.”
The latest revelations come at an awkward time, with Lady Davidson and Mr Ross due to campaign together in Edinburgh on Wednesday ahead of the local elections.
They had hoped to send a message of unity to pro-UK voters to persuade them to back the Tories next month, but will now face a barrage of questions over partygate.
The row also threatens to overshadow the launch of the Scottish Conservatives’ local election manifesto later this week.
Both Mr Johnson and Chancellor Rishi Sunak were found as part of a police probe into allegations of lockdown parties held at Downing Street.
Carrie Johnson, the wife of the Prime Minister, also received a fixed-penalty notice.
The fines are among more than 50 handed out so far on the back of Operation Hillman, which is looking into breaches of Covid regulations at the top of government.
Opposition parties have called for Mr Johnson to resign and for Westminster to be recalled from recess.
The Prime Minister said he had paid the fine and offered a “full apology”.
In a statement, Mr Ross said: “The public are rightly furious at what happened in Downing Street during the pandemic.
“I understand why they are angry and share their fury. The behavior was unacceptable.
“The Prime Minister now needs to respond to these purposes being issued.
“However, as I’ve made very clear, in the middle of war in Europe, when Vladimir Putin is committing war crimes and the UK is Ukraine’s biggest ally, as President Zelensky said at the weekend, it wouldn’t be right to remove the Prime Minister at this time.
“It would destabilize the UK Government when we need to be united in the face of Russian aggression and the murdering of innocent Ukrainians.”
Asked if Mr Johnson should lead the Tories into the next general election, Mr Ross was evasive.
He told Sky News: “We don’t know where we’ll be in a couple of years’ time, when the next general election is due.
“I’m looking at the situation right now. I’m looking at where we were just a few months ago, when I previously spoke out against the Prime Minister, and I’m looking at where we are now.
“Things change a lot in a very short period of time.”
He said “every single person who has received a fixed-penalty notice has to apologize for that and explain their actions”.
Last month, Mr Ross said “the middle of an international crisis is not the time to be discussing resignations”.
He had previously led calls for Mr Johnson to resign and was backed by almost all of his MSPs in Holyrood.
Deputy First Minister John Swinney, who is also Scotland’s Covid Recovery Secretary, was among those to criticize Mr Ross’s position.
Senior SNP MP Stewart McDonald said Mr Ross had “forfeited the right to be taken seriously”, branding him “a thoroughly ridiculous figure”.
He tweeted: “The PM has indeed been a strong ally of Ukraine – I have given him credit for it – but that does not entitle him or his party to use Russian war crimes and the victims of them as a shield for his own lying and lawbreaking. The mere suggestion that he might is a new low.”
He accused Mr Ross of “using the war in Ukraine for domestic political purposes”, adding: “That’s the opposite of what we all agreed was decent and right. It’s revolting.”
Scottish Labor deputy leader Jackie Baillie said: “This is a shameful attempt by Douglas Ross to hide his lack of principle behind the ongoing tragedy in Ukraine.
“That Douglas Ross does not have the spine to condemn Boris Johnson’s actions and demand his removal from office is simply shocking.
“Even Ruth Davidson – who is currently being wheeled out by Scottish Tory spin doctors to save them from electoral oblivion – has said that Boris Johnson must go.
“No one is above the law and it is simply unacceptable to use the human tragedy in Ukraine to shield the Prime Minister.
“Douglas Ross should hang his head in shame.”
The Scottish Liberal Democrats accused Mr Ross of having “the backbone of a jellyfish”.
The party tweeted: “While women gave birth alone and people said goodbye to loved ones on Zoom, the Prime Minister and the Chancellor were partying in Downing Street.
“Douglas Ross needs to show some guts and leadership and call for their resignations.”