In a statement read out by his solicitor Aamer Anwar on the court steps, Siddique said: "I have always maintained my innocence, but they took my liberty, destroyed my family's reputation and labelled me a terrorist.
"But I never had any bombs or plans to hurt anyone. In court it was said I was a wannabe suicide bomber, but I have always said I was simply looking for answers on the internet."
The shopkeeper's son was jailed for eight years in October 2007 after a four-week trial in Glasgow.
He was found guilty of two charges under the Terrorism Act 2000, one under the Terrorism Act 2006 and a breach of the peace.
The most serious charge related to the possession of articles that gave rise to "reasonable suspicion" they were connected to terrorism.
His conviction on that allegation resulted in a six-year prison term.
But at his appeal hearing in January Lord Osborne criticised the way the trial judge explained the main Terrorist Act charge to the jury.
The judge, sitting with Lords Reed and Clarke in Edinburgh, said the "material misdirection" amounted to "a miscarriage of justice".
In a statement, the Crown Office said Siddique remained convicted of serious terrorist offences.
The offences relate to the setting up of websites which provided links to documents on how to operate explosives and weapons, and to circulating terror publications via the web.
He is also still guilty of a breach of the peace for showing images of suicide bombers, murders and beheadings to fellow students.
The Crown Office added: "After careful consideration of the Appeal Court's judgment, the Crown has decided not to seek authority for a retrial.
"The fact that Mr Siddique has already served the majority of his sentence for charge one, and has de facto served his sentence in full for the other terrorist offences, of which he remains convicted, means that a retrial would have little practical effect.
"Accordingly, the Crown has concluded that a retrial would not be in the public interest."
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