Richard and Margaret Hewat of Hewats on the Mile, which describes itself as one of Edinburgh's top restaurants, posted the letter outside their premises on the Royal Mile.
It comes after Ms Sturgeon extended the closure of pubs and restaurants in the Central Belt ahead of the introduction of a new five-tier virus alert system in early November.
The Scottish hospitality sector has warned that many outlets "cannot survive" if restrictions remain in place indefinitely.
The open letter from Hewats, which serves modern French and Scottish cuisine, said it closed on March 16 at a “huge” financial cost to the business and with the landlord still wanting rent.
The restaurant later reopened at a cost to get the premises Covid-compliant, with its tables reduced from ten to five.
"We are a small, family-run restaurant and had no-shows, late cancellations, and then you introduced the 10pm curfew, stopping our second sitting on Fridays and Saturdays, denting the income we could generate,” the open letter said.
The restaurant – which started out in 2004, moving from Causewayside to its current location in 2017 – still had to pay bills for utilities, rent, a slice of furlough, insurance and extra cleaning.
“Then, with two days’ notice, out of the blue you want us to close,” the owners added.
While the owners had to close the premises on October 9, they had to wait until the 20th “to apply for scraps”.
The owners said they had “not seen a penny throughout the closure period” while other businesses, such as full cafes with no physical distancing, fast food chain McDonald’s and supermarkets, have been busy.
Hewats’ open letter added: “Yes, you are doing a difficult job in unprecedented times, but where you have lost me is closing restaurants and not having funds in place after four months’ full lockdown… Our cashflow is empty!”
Meanwhile, Eusebi Deli in Glasgow’s West End has won a legal battle to reopen as a cafe, despite the tough restrictions in the city.
Lawyers acting for the business secured an interim interdict on 19 October, which will prevent Glasgow City Council from closing the restaurant, as the premises meets the legal definition of “café”.
All licensed premises in the Greater Glasgow and Clyde, Lanarkshire, Ayrshire and Arran, Lothian and Forth Valley health board areas remain closed for both indoor and outdoor operations.
Cafes without a licence to sell alcohol will be allowed to open until 6pm, Nicola Sturgeon said, to counter social isolation.
Meanwhile, Edinburgh’s One20 Wine Cafe has won a court battle with the council after being closed for serving food “too smart and too fancy” to be considered a cafe. Restaurants in the Scottish capital to have closed for good since the onset of the pandemic include Le Roi Fou and Superico 83.