He went on to denounce former colonial power Britain, as well as French President Nicolas Sarkozy and US President George W Bush, who both called earlier this week for the 84-year-old to resign.
"Because of cholera, Mr Brown, Mr Sarkozy and Mr Bush want military intervention," Mr Mugabe said. He added: "Let's tell them that the cholera cause doesn't exist any more."
He was speaking in the capital, Harare, at a state funeral for senior ruling party official Elliot Manyika, who died in a car crash over the weekend.
Shortly after Mr Mugabe spoke, the UN Office for the Co-ordination of Humanitarian Affairs said the toll from the disease had risen slightly overnight to 783 and that 16,403 were believed to have been infected.
But Save the Children in Harare said these figures were an underestimate and the epidemic was almost certainly worsening.
The agency's Rachel Pounds said on Thursday: "Reliable figures are hard to come by, but there is much evidence out there that this crisis is growing, not diminishing."
She added: "With even the most basic healthcare on hand, you would expect to see a death rate of only one or two per cent. In some areas of Zimbabwe a third of those who have contracted the infection are dying."
New footage from Harare shows open sewers and residents drawing water from wells
The WHO has warned that the total number of cases could reach 60,000 unless the epidemic was stopped.
Hundreds of Zimbabweans have crossed into South Africa to seek medical treatment because Zimbabwe's health service and water supply infrastructure have virtually collapsed.
At least eight people have died in Limpopo Province, prompting the authorities to declared the Vhembe district which includes the border town of Musina a disaster area.
President Mugabe's ruling Zanu-PF party and the opposition Movement for Democratic Change (MDC) have been deadlocked in power-sharing negotiations for several months.
Mr Mugabe also said in Thursday's speech: "Shall we also say that [because] there is mad cow disease, there must be war, Britain must be invaded. Mr Brown, your head must go for some medical correction."
"We are not a threat to international peace, not a threat to our region," he added.
There has been growing international pressure recently for Mr Mugabe to step down.
Cholera has spread because of Zimbabwe's failing health system
The UK has led calls for Mr Mugabe to go. African countries like Botswana and Kenya have also said he should step down, but South Africa has refused to call on Mr Mugabe to quit.
And the 53-member African Union said on Tuesday the only solution to Zimbabwe's crisis was the power-sharing talks.
On Monday, European Union nations ramped up the diplomatic pressure on Zimbabwe's government, broadening sanctions on President Mugabe and his inner circle.
Mr Mugabe and MDC leader Morgan Tsvangirai signed a power-sharing deal in September but talks on forming a unity government have stalled as both sides bicker over how to divide control of key ministries.
South Africa, led by former President Thabo Mbeki, has been mediating the negotiations but the MDC has accused him of not putting enough pressure on Mr Mugabe to share power.
The MDC seized a majority in parliament in March elections, when the opposition leader defeated Mr Mugabe in a first round presidential vote.
Mr Tsvangirai pulled out of a June run-off, accusing Mr Mugabe's party of orchestrating a wave of political violence.
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