The shock demand came after the government was accused of covering up the failed test which saw the launch of an unarmed Trident II D5 missile from a British submarine off the coast of Florida in June.
On the BBC’s Andrew Marr show on Sunday, the PM repeatedly refused to deny she knew about the malfunction when addressing MPs before a crucial Commons vote on the future of the submarine-launched missile system last year.
Mrs May repeated refusal to respond did not sit well with the LBC caller as he called on her to stand down.
Less than impressed with Ed from Hounslow's request, Ferrari asked him: “Just to prove how out of touch you are…. who was Prime Minister in June?
LBC • BBC
Ferrari shut down the caller who demanded Theresa May should quit with one simple fact
"It was David Cameron. There's every possibility that Mr Cameron knew and not Mrs May. So why the hell should she go?"
Refusing to back down from the argument Mrs May should quit he continued: "I'll tell you why because when we vote in a democracy for somebody to lead our country, we expect them to be open, honest and truthful."
To which the LBC host brilliantly retorted: "Which of the words are you struggling with? Let me try again.
“It is possible that it's Mr Cameron who was informed because he was Prime Minister back in June."
There's every possibility that Mr Cameron knew and not Mrs May
He continued: “Now we look at the cover up but if it was actually David Cameron who initially knew why you say Mrs May has to go?"
During the BBC interview on Sunday, Marr grilled the PM after she repeatedly failed to answer direct questions about her knowledge of the test ahead of the crucial Commons vote.
Mrs May said: "I have absolute faith in our Trident missiles.
10 facts about Trident nuclear weapons
Wed, July 20, 2016
As MPs vote to renew the UK's Trident weapons system, we look at the facts about the nuclear weapon.
1 of 10
The UK's Vanguard fleet of four submarines carrying Trident nuclear missiles are due to become obsolete by the end of 2020
"When I made that speech in the House of Commons, what we were talking about was whether or not we should renew our Trident, whether or not we should have Trident missiles, an independent nuclear deterrent in the future.
"I think we should defend our country, I think we should play our role in Nato with an independent nuclear deterrent.”
When Marr asked about her knowledge of the test for a final time, Mrs May said: "There were tests that take place all the time for our, regularly, for our nuclear deterrents.
"What we were talking about in that debate that took place was about the future."