Lawyers representing the man mounted a High Court challenge after a judge at Westminster Magistrates’ Court had ruled that he should be extradited.
They said the man had mental and physical heath difficulties and argued that extradition would be oppressive and unfair.
But a judge dismissed the man’s challenge on Wednesday.
Mr Justice Chamberlain had considered the case at a High Court hearing in London earlier this month.
Lawyers representing the Irish state had said the man’s appeal should be dismissed.
The judge heard that the man had been accused of committing rape in Waterford in 2001 and prosecutors wanted him to appear in court in Ireland.
He was told the man, who lives in supported accommodation in Essex, had been accused of rape after police reviewed the case.
Mr Justice Chamberlain said the woman could not be identified in media reports of the case.