Annie Borjesson's body was discovered on Prestwick Beach on December 4, 2005, and almost nine years on her family continue to look for answers.
They believe the 30-year-old was murdered, but the initial police inquiry found she had drowned and there were no suspicious circumstances.
Police Scotland has now reviewed the evidence in the case but claims there is nothing to contradict the original findings.
The decision has been described as "shameful" by campaigners acting on behalf of Ms Borjesson's family.
Kenneth Roy, editor of online magazine Scottish Review, which has long supported the family, said he had written to Ms Borjesson's relatives to "let them know that we had done our best but that our best had not been good enough".
He added: "I had to say that, barring some wholly unexpected development, I could see no way forward. The case is closed. Shame on Scotland."
The call for a review of the evidence was triggered by a dossier that was compiled by Scottish Review after the family instructed their own investigations in the case.
The Swedish forensic service is reported to have carried out tests on Ms Borjesson's bone marrow, which discovered that algae found in fresh water and not sea water was in her bone marrow.
Mr Roy claims that according to marine experts, the presence of such algae does not support drowning as the cause of death.
He said: "There would have been - and remains - a way of resolving the doubts. Other body tissues could have been released for expert examination and the results made public with the family's permission.
"The authorities have steadfastly refused to allow this. What do they have to fear?"
The editor added: "When Annie's body was returned home to Sweden after nine days, the undertakers there were taken aback to discover extensive areas of bruising which had not been recorded in the autopsy report and which, in their authoritative opinion, could not have been caused by post-mortem lividity.
"No explanation has ever been offered for the divergence between the undertakers' observations and the autopsy report.
"We have the undertakers' signed statement. We would have been happy to make it available to Police Scotland. But we were never asked for it. How, in the face of so much unexamined evidence, can [Police Scotland] maintain that there is nothing which contradicts the findings of the initial investigation?"
Ms Borjesson's family suspect she was murdered elsewhere and that her body was dumped on the beach.
They are suspicious of the location of her body and have said there are experts willing to look at tidal records from the time to determine whether it was at all likely that the body could have washed up in that position.
However, Mr Roy claims Police Scotland has ignored this suggestion.
A letter from the force about the review states: "This review is now complete and I am entirely satisfied that there is nothing within the publication which contradicts the information which was provided to [the Crown Office] during the initial investigation."
Assistant Chief Constable Malcolm Graham, who conducted the review, has offered to meet the family if they wish to discuss it further.
Police Scotland was unable to comment on the case.