The ITV star issued the ultimatum as pressure mounts on Number 10 over claims the top aide to the Prime Minister breached lockdown rules.
Mr Cummings stands accused of breaking Government rules by travelling hundreds of miles to his parents’ home in County Durham from his home in London while he and his wife were self-isolating due to suspected Covid-19 symptoms.
Downing Street has defended the 48-year-old strategist, insisting his actions “were in line with coronavirus guidelines” and denying reports he was spoken to by the police.
Cabinet minister Michael Gove has also leapt to his defence, tweeting: "Caring for your wife and child is not a crime."
But in a tweet liked thousands of times, Mr Morgan said on Saturday morning: "If Boris doesn’t fire Cummings today, then I will deem the Lockdown over & drive down to see my parents (from a 2m distance) for the first time in 12 weeks.
“I’m not having one rule for these clowns & another for the rest of us.”
He also called on the PM to front today's Downing Street briefing "or be damned by your cowardice".
Political leaders have piled pressure on Mr Johnson to sack the aide, with the Lib Dems branding it “outrageous” and Labour saying the “country deserves answers”.
Labour said in a statement: “We are still waiting for a clear explanation from No.10 about Dominic Cummings’ actions
“The public have made huge sacrifices during lockdown. It can’t be one rule for those who set them and another for the British people. The country deserves answers at today’s press conference.”
Lib Dem leader Ed Davey said: “People have not seen their loved ones, they've had to cancel weddings, they've not been able to go to the funerals of their closest relatives.
“It looks quite outrageous that the Prime Minister's closest adviser has done this, and frankly if that turns out to be the case he must go... he should either resign, or the Prime Minister should sack him.”
SNP Westminster leader Ian Blackford called Mr Cummings’ alleged actions the “height of irresponsibility” and said he “should have gone” when the story broke last night.
Former Conservative MP David Lidington, who was de facto deputy PM under Theresa May, was also among those saying the news raised serious questions.
“There’s clearly serious questions that No 10 are going to have to address not least because the readiness of members of the public to follow government guidance more generally is going to be affected by this sort of story,” he told BBC Newsnight
A Downing Street spokesman said: “Owing to his wife being infected with suspected coronavirus and the high likelihood that he himself would become unwell, it was essential for Dominic Cummings to ensure his young child could be properly cared for.
"His sister and nieces had volunteered to help so he went to a house near to but separate from his extended family in case their help was needed. His sister shopped for the family and left everything outside.
"At no stage was he or his family spoken to by the police about this matter, as is being reported.
“His actions were in line with coronavirus guidelines. Mr Cummings believes he behaved reasonably and legally.”