Almost 15,000 lent their signatures to the petition, set up by campaigning group Republic, which is also calling on the Government to publish a report into much the event will cost the public purse.
"A royal wedding is a private, personal event, dressed up as a national occasion,” states the petition, which was launched last month. “That lets the royals use the wedding as a PR exercise and to expect the taxpayer will pay a large part of the costs.”
“The exact details of royal wedding funding are shrouded in secrecy, but we do know - at the very least - that expensive road closures and policing will be required. And we know local councils and the taxpayer will end up footing the bill.
“Taxpayers should not be funding a private wedding, no matter who is getting married.”
Prince Harry and Ms Markle are set to marry at St George’s Chapel, in Windsor Castle – where the Prince was christened – on 19 May.
Kensington Palace announced earlier this month 1,200 members of the public will be among a crowd of 2,640 allowed into the castle grounds to witness the wedding day.
It said the royal family would pay for the celebrations, though that is unlikely to include security costs.
For the 2011 marriage of Prince William and Kate Middleton, the taxpayer was estimated to have contributed as much as £20m for thousands of additional police officers.
“If Harry and Meghan want to turn their big day into a public event, they need to pick up the bill – all of it,” the Republic petition continues.
“The Palace claims the wedding will be funded by the royal family, but royal funding blurs the lines between private income and public money. So, whether it's the cost of policing paid for directly by us, or costs of the wedding ceremony, paid for by the royals, the taxpayer still ends up paying.”
Graham Smith, chief executive of Republic, told The Independent the petition’s success showed anger was “greater than ever” over public spending on the royal family at a time of “squeezed budgets”.
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