The theatre impresario said the current system is “completely, completely untenable” and his industry has been left “on its knees”.
He had earlier announced Cinderella would not be returning to the stage on Monday after a member of the cast tested positive for Covid-19 on Saturday.
While the rest of the production tested negative, Lord Lloyd-Webber said “‘freedom day’ has turned into closure day” as a result of the isolation rules.
Speaking at the Gillian Lynne Theatre on Monday, the composer said “we cannot function with this current system”.
He said: “We can’t isolate every time somebody may or may not have it. It just simply doesn’t work.
“I say this from the heart, I am seeing the profession I have loved, I’m seeing musical theatre I think I had a small part in pioneering in this country at a time when frankly the British were not considered to be people who could do musicals.
“And I am just saying it with passion – please, please will this Government for once listen to us.
“Listen. We do know what we’re doing, we do. Just listen and knock all these platitudes and endless, endless blunt instruments that don’t apply across the board.”
An emotional Lord Lloyd-Webber said
“I have tried and I have tried and I have tried” to work with the Government and prove theatres are safe.
He said the Gillian Lynne, where Cinderella was due to open, has “100% fresh air” and the “best ventilation system you can find”.
Lord Lloyd-Webber warned he had heard of another “very, very major show” that will no longer be opening on schedule.
He added: “We can’t go on like this, theatre is now on its knees, there’s no way forward.”
He said he does not know how long it will be until the industry is back to normal, adding Prime Minister Boris Johnson “doesn’t seem to grasp what we’re about in theatre”.
Lloyd Lord-Webber said he is determined to open Cinderella in London and has so far ignored “siren voices” suggesting he moves it to Broadway.
“I’m not going to do that, I can’t do that,” he said. “I’m a Brit and we’ve got a fantastic cast and so we will open here, but who knows? 2084? I don’t know.”
He was critical of the Government’s overall messaging, accusing ministers of “never really doing the research”.
He said the Government must allow the theatre industry to prove they should be allowed to proceed, saying it is “no good” waiting until mid-August, when a testing regime will replace the requirement for fully-vaccinated contacts to isolate.
“I do worry at the end of the day that the Government doesn’t regard theatre as anything other than nice to have,”
“I don’t think they have a clue of what the real economic value to the country theatre is, and indeed all forms of live entertainment.”
He outlined the financial boost the theatre industry brings the country and said: “It’s so vital this Government understands the economics of what they’re doing, let alone the idiocy.”
Asked about seeing thousands of fans at Wembley for the Euro 2020 football tournament and crowds of people at nightclubs on Monday, Lord Lloyd-Webber said:
“The fact of the matter is we are a law-abiding lot. For us to be lumped together, as we have been, with certain other sectors seems to be grossly wrong and unfair.”
He said Culture Secretary Oliver Dowden “does his best” but “I don’t think he’s really able to be that effective”.
“This is the last chance for this Government to show they have the remotest interest in theatre,” Lord Lloyd-Webber said.
He added: “There comes a time, I suppose, where we simply won’t be able to go on. There does come a time. We really are pretty much at that point now. I am determined to keep it going for as long as we can but there does come a point where we can’t.”
Earlier, Mr Dowden said he was “deeply disappointed” to hear about the cancellation of Cinderella.
He tweeted: “I know how much effort ALW has put into this production, and how much it meant to the cast, crew and theatre lovers alike.
“Whilst the need to self isolate is an economy wide issue, I recognise the particular challenges it presents to the arts and I’m strongly making the case for that in government.
“In the meantime, I hope to see as much open as possible and, where shows are restarting, please give your support by buying a ticket.”
Last week, actors’ union Equity called for self-isolation rules for the entertainment industry to be changed, warning current regulations are having a “devastating and costly impact”.
A Government spokeswoman previously defended the support given to the arts, saying the sector has been given “unprecedented” backing.
Cinderella opened last month with an audience capacity of 50% after the impresario rejected Mr Johnson’s offer for the show to be included in the live events pilot scheme.
Lord Lloyd-Webber previously said proceeds from Monday’s show would be donated to St John Ambulance and the NHS.
Written by The Crown star Emerald Fennell and starring Carrie Hope Fletcher, Cinderella is described as a “complete reinvention” of the classic fairytale and is based on an original idea by Fennell.