The footage, showing his girlfriend’s pug raising its paw in response to statements including “sieg heil” and “gas the Jews”, was viewed more than 3 million times on YouTube.
Meechan attempted to overturn his conviction but was refused leave to appeal by Scotland’s appeal court last August, then tried to take the case to the UK Supreme Court.
Three judges – Lord Reed, Lady Black and Lord Kitchin – refused to hear Meechan’s case.
Official documents said they gave the reasons as: “The application is incompetent, and in any event without merit.”
The decision was made on 6 November, but Meechan did not tell supporters who donated almost £196,000 for the case until Thursday.
He said no further information on why the application was refused had been given and suggested he could appeal to the European Court of Human Rights (ECHR).
“We are currently seeking advice from an expert on the ECHR to see whether or not this route would be worth our time,” Meechan wrote. “I do not wish to invest a lot of time and money into what could turn out to be another very expensive ‘no’.”
He told supporters that funds remaining after legal fees would be donated to charity.
During the first hearing at Airdrie Sheriff Court, Meechan claimed the video was intended as a joke to annoy his girlfriend and did not reflect his views.
He then attempted to overturn both his conviction and sentence, but was refused leave to appeal by the Sheriff Appeal Court in Edinburgh in August 2018.
A letter from the court said the appeal was “without merit … not arguable and in each of its elements is wholly misconceived”.
“This was a deeply unpleasant offence in which disgraceful and utterly offensive material was very widely distributed by the appellant,” it said.
“This was to the considerable distress of the community in question and – just as disturbingly – to the apparent approval of a large number of persons who appear to share the appellant’s racist views.”
Meechan, a self-described “professional s**tposter”, has denied accusations of Nazism and racism and claims to campaign for freedom of speech.
The case provoked widespread concern from free speech campaigners and comedians, and became a cause celebre for right-wing extremists.
Tommy Robinson was among supporters at a sentencing hearing, and Meechan later joined Ukip under anti-Islam leader Gerard Batten.
He was listed as a candidate for the party in May’s European parliament elections, but was not elected and announced he had left the party last month.