The former British ambassador to Uzbekistan was sentenced to eight months in prison for breaching a court order protecting the identities of women who accused former first minister Alex Salmond of sexual assault.
He applied to the High Court seeking permission to appeal to the UK's highest court against the verdict of contempt of court and the sentence.
But in a written judgement issued on Tuesday, Lord Justice Clerk Lady Dorrian refused the application, stating there were “no arguable points of law arising”.
In a 13-page judgement, published online, she said the contempt of court order did not prevent reporting of Mr Salmond’s trial, or any other matter of public interest, but was in place to protect the identification of the complainers.
It is understood that Murray, 62, now plans to apply directly to the Supreme Court to hear the appeal.
The suspension of his committal warrant has been continued for a further four weeks in the meantime.
In her judgement, Lady Dorrian noted Murray described himself as a “journalist in new media”.
She said that “whatever that may involve,” it was relevant to distinguish his position from that of the mainstream press, which is regulated and subject to codes of practice and ethics “in a way in which those writing as applicant does are not”.
Murray’s legal team had argued his sentence of eight months’ imprisonment was disproportionate and incompatible with article 10 of the European Convention on Human Rights.
But the High Court judgement ruled the actions of Murray “were such as struck at the heart of the fair administration of justice”.
It went on to explain: “The suggestion is made that the applicant’s genuinely held belief that the prosecution of the former first minister was unwarranted is the relevant matter of public interest, the inference from the context of that submission being that the sanction is such as
to prevent discussion of a legitimate matter of public concern.
"However, that is not a tenable argument. It is the repeated publication of material likely to lead to identification of complainers in the face of a clear order of the court prohibiting that which drew the sanction.
"The order did not prevent discussion of whether the prosecution was objectively justified. The applicant remains free to pursue discussion of that issue, as long as the anonymity of the complainers is respected.”
At the time of Murray’s sentencing at the High Court in Edinburgh last month, Lady Dorrian said his repeated breaches, which included refusing to remove blog posts despite legal warnings from the Crown Office, represented a "contempt of considerable gravity".
Murray's offending posts were written over a period of a month and remained up, unredacted, despite him being told they could potentially lead to the identification of women who had made complaints about Mr Salmond, who was eventually acquitted of all 13 charges.
Lady Dorrian explained: "It appears from the posts and articles that he was in fact relishing the task he set himself, which was essentially to allow the identities of complainers to be discerned – which he thought was in the public interest – in a way which did not attract sanction."