Nicolas Sagar was working as a pianist on the island and had gone swimming. It is believed he got in to difficulties and drowned.
The 33-year-old was found at Maho Beach by tourists who could not resuscitate him and called emergency services.
Medics pronounced him dead at the scene on Sunday morning and his body was identified later that day.
Mr Sagar's parents are working with the Home Office to repatriate their son's body and arrangements are still being made for his funeral.
Police on the island said there appeared to be no suspicious circumstances and that the incident was a tragic accident.
Mr Sagar, who was also involved in acting and musical directing, previously worked in musical theatre in London and on cruise ships.
He was expected to continue his travelling by visiting Barbados on Monday.
'Regular swims'
Kelly Sagar, the deceased's father, said: "He was a wonderful son. The piano was his great love; from the age of eight, the only thing he wanted to do was play piano and sing.
"He was never happier than when he was sitting at the piano with a pint resting on top.
"He was travelling the world doing what he loved.
"Nick worked in the evenings and liked to spend his days on the beach. He'd go swimming there quite often and it appears that on this occasion he got in to difficulty.
"He will be missed terribly."
Mr and Mrs Sagar said they had been comforted by messages posted on their son's social networking site.
A musician from Alloa has died after drowning off the coast of the Caribbean island of St Maartens.