The GMB presenter was reacting to the Prime Minister's apparent confusion over Covid-19 restrictions.
"He not only doesn't know the rules, he also breaks the rules that he does know", Morgan said.
The breakfast show played a clip of Mr Johnson attending a woodwork class during a public visit to Exeter college.
The Prime Minister did not wear a mask as he launched his new plan to retrain Britain.
"People around him thinking nothing other than who is this drongo, and what is he doing", Morgan added.
"He couldn't even shave wood properly. He is just a bumbling fool. It has got beyond a joke.
"This man is running the country in the biggest crisis, health-wise and economically, that we have faced in our lifetime and he is a bumbling fool who doesn't even know his rule of six.
"Something has got to change. We cannot be in this rudderless position with the captain of the Titanic taking us down."
It came after Mr Johnson was forced to apologise for getting advice on restrictions in north-east England wrong.
The Prime Minister stumbled over his explanation of the ban on households mixing, just hours after another member of his Government was unable to say whether the prohibition extended to pub beer gardens.
Mr Johnson said he “misspoke” but the confusion comes as the Government is under pressure from Tory rebels to give MPs a greater role in scrutinising and voting on new measures before they are implemented.
The new regulations in Northumberland, Newcastle, Gateshead, North and South Tyneside, Sunderland and County Durham will see fines imposed if people mix with others from outside their household or bubble in indoor settings, including pubs and restaurants.
But with the rules yet to be published in full, ministers faced questions about whether they would apply in settings such as beer gardens.
Education Minister Gillian Keegan was unable to answer on BBC Radio 4’s Today programme and Mr Johnson added to the confusion.
“In the North East and other areas where extra-tight measures have been brought in, you should follow the guidance of local authorities but it is six in a home, six in hospitality but, as I understand it, not six outside,” Mr Johnson said in response to questions following his address to the nation.
But he later tweeted: “Apologies, I misspoke today. In the North East, new rules mean you cannot meet people from different households in social settings indoors, including in pubs, restaurants and your home. You should also avoid socialising with other households outside.”