Villa Taranto, on the shores of Lake Maggiore in the north of the country, was created by Capt Neil McEacharn in the 1930s.
Around a third of its trees and exotic shrubs were damaged when it was hit by a freak storm packing winds of up to 70mph at the weekend.
The managers of the historic garden, which attracts around 160,000 visitors a year, say they are in desperate need of funds to repair the damage.
They have called for the government to declare the area a state of emergency so that funds can be released for its restoration.
Massimo Nobili, the president of the surrounding province, went a step further, appealing for international donors to send donations.
He said the garden was “a prominent tourist attraction” that was vital for the local economy.
The 40 acre garden, rated as one of the most important in Europe, contains around 20,000 species of plant, including a type of sacred tree from India, Japanese maples and giant Amazonian lilies.
The estate was bought in 1931 by Capt McEachern, an Eton- and Oxford-educated Scot who was the son of a shipping magnate.
The guardians of the site said the storm was a “calamity” which had forced the indefinite closure of the garden.
After weeks of stifling heat, much of Italy was hit by wild weather at the weekend, including storms and torrential rain.
Capt McEachern named the garden after one of his ancestors, a marshal in the French army who was given the title Duke of Taranto by Napoleon.
After spending nine years creating the garden, the Briton bequeathed it to the Italian state, which opened the property to the public in 1952.