Human rights groups have voiced concern at the short time it has taken for two Thai fishermen to be sentenced to death for killing student Katherine Horton.
Amnesty International and Fair Trials Abroad have questioned the fairness of the trial, completed in three weeks.
Wichai Somkhaoyai, 24, and Bualoi Posit, 23, now face execution for the murder of Katherine Horton, who was 21.
But Conservative MP Stephen Crabb said he respected Thailand's right to deal with the men by its own processes.
The fishermen were sentenced to death on Wednesday, two-and-a-half weeks after Miss Horton's body was found in the sea off the island of Koh Samui on 2 January.
Lawyers for the men immediately said they would appeal.
Miss Horton's family, who on Tuesday led hundreds of mourners at her funeral, issued a statement thanking the Thai authorities for their speed in finding the men responsible.
The statement made clear that, contrary to reports, the family had not expressed an opinion about what should happen to the men.
Katherine Horton's family have thanked the Thai police
But Amnesty International campaigner Katherine Gerson told BBC Radio Wales the organisation "regrets" the imposition of the death penalty.
"We understand that the judge has stated he imposed the death sentence because of the inhumanity of the crime," she said.
"We would say that it is an inhumane treatment and would urge the authorities not to execute the individuals who have been sentenced."
She said the case raised "many questions" about death penalty trials in Thailand, and queried why a case with that "level of priority" was conducted at such speed.
"If Thailand executes these two men, it will be a step backward for a country which has been moving away from the death penalty," she said.
Disagreeing with comments by the Thai prime minister ahead of the trial which called for the death sentence, she added: " That is for the judiciary to decide."
Fair Trials Abroad said they were "disappointed" by the sentence.
Spokesman Stephen Jacobi said he was not aware of a death sentence being imposed in similar murder cases in Thailand and highlighted the jail term given to a police sergeant who killed a British couple last year.
But the Conservative MP for Preseli Pembrokeshire, Stephen Crabb, said while he did not support the death penalty in the UK, every country had its own judicial processes.
"This has been dealt with by the Thai authorities and I respect their right to proceed according to the established judicial processes in their country," he said.
"I don't suspect many people in the UK will have any sympathy with these men. They have pleaded guilty to a brutal and disgusting crime."
Conservative Monmouth MP and Welsh assembly member David Davies said: "Rather than thinking about these two rapists and murderers my thoughts shall be with the parents and family and friends of the poor girl who was murdered."