Leading Seaman Timothy Andrew MacColl, 27, from Gosport in Hampshire, was last seen getting into a taxi at 02:00 local time on Sunday after a night out.
The ship had arrived in Dubai on Saturday and left on Thursday.
The family of the father-of-two, who is originally from Killin, Stirlingshire, said they were "completely at a loss and going crazy with worry".
Leading Seaman MacColl, known as Timmy to family and friends, was last seen by a shipmate getting into a taxi alone.
He had just left the Rock Bottom Cafe at the Regent Palace Hotel, in the Bur Dubai area, which is about a 15-minute drive from Port Rashid where his ship was docked.
His wife Rachel, who is expecting the couple's third child in October, said there had been no sightings of her husband and that checks with hospitals, prisons, police stations and mortuaries had so far found no record of him.
"I last spoke with Timmy at midday on Saturday and he was in really good spirits and looking forward to speaking with the children on Skype on Sunday," she said.
That call never happened, she said, adding that it was "completely out of character" for him not to contact his family.
"We are appealing to the people of Dubai and reaching out to the large expat community to help us find Timmy and raise awareness in that area," she said.
"Wherever Timmy is, I know that he wants to be found, he will want us to find him, we need people to help us do that."
The couple's two children are aged six and four.
Leading Seaman MacColl is described as white, 5ft 8ins tall, of medium build, with short brown hair cropped with a flicked fringe, and speaks with a broad Scottish accent.
At the time of his disappearance, he was wearing a red MotoGP T-shirt, dark-coloured long shorts and white trainers.
A spokesman for the Royal Navy said the service was working with local authorities and British Foreign Office staff to try to find the missing sailor, and providing support to his family, he said.
"Everything is being done to ensure that he is located safely," the navy spokesman said.

The staff at Brickkiln Waste Limited on Heather Road were told on Friday that the company was going into administration.
It is understood Brickkiln has been involved in a dispute with Derry City and Strabane District Council over payment for contracts.
Brickkiln recently had its licence temporarily suspended by the Northern Ireland Environment Agency.
This was because of breaches at its waste site in Maydown.
In a statement, Derry City and Strabane District Council said that following the Northern Ireland Environment Agency's (NIEA) suspension of the company's waste acceptance services on 10 July, it had been working closely with Brickkiln, the NIEA and other councils to "assist in resolving ongoing waste processing issues".
"Prior to this cessation of service, council was forced to suspend payment to the company as Brickkiln had failed to fulfil contractual obligations as set down in contract specification," the statement added.
"Where invoices have been received by council demonstrating that the company has met their agreed terms, these invoices have been paid in full.
"Council is legally obliged to adhere to all legislation in terms of the appropriate management of waste and is committed to working with the company, the NIEA, and other councils to resolve the matter."
Earlier this month, 10 appliances and almost 50 firefighters were involved in tackling a fire at the Heather Road site.
A British sailor has gone missing in Dubai while his ship, HMS Westminster, was docked, the Royal Navy has said.
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Fifty workers at a Londonderry waste firm have lost their jobs.