Mark Meechan, 32, who uses the moniker Count Dankula, was prosecuted after teaching the dog, a pug named Buddha, to perform the offensive gesture.
He was convicted of a hate crime in communicating a video that was "grossly offensive" after denying the charge at a trial at Airdrie Sheriff Court.
Meechan, from Coatbridge in North Lanarkshire, taught his girlfriend's pug to react to the words "gas the Jews" and filmed it for his YouTube channel in 2017, supposedly to annoy his girlfriend.
The pug was also seen raising a paw to fascist chant "Sieg Heil" during the footage called "M8 Yur Dug's a Nazi", which was viewed more than three million times.
The case sparked a debate around freedom of speech with comedians such as Ricky Gervais speaking out on Meechan's behalf, as well as attracting the attention of far-right leader Tommy Robinson who attended the trial in support.
Some £195,657 was raised on a GoFundMe page to take the appeal to the Supreme Court, which rejected the case saying it had 'no merit', according to Meechan.
He wrote: "As some of you may be aware, we petitioned to the UK Supreme Court to hear our case, a process that took very many months to hear back from and sadly we have recieved news back that they have refused to hear the case.
"The reason they cited is that my appeal "had no merit" but as of yet, they have provided no information as to specifically why this is, I wish I had further information to give you as to specifically why they rejected the appeal, but sadly, they have provided no further explanation.
"After this step we have one final option, the European Court Of Human Rights.
"This process would again be very time and fund consuming and at this point we are only considering going down this path, as in my personal view from freedom of speech cases I have seen handled by the ECHR in the past, they do not have the best track record.
"Again, that is my personal view.
"We are currently seeking advice from an expert on the ECHR to see whether or not this route would be worth our time, I do not wish to invest a lot of time and money into what could turn out to be another very expensive "no" with again, no proper explanation given, when the funds used for it could have instead gone to a charity to help those in need.
"One hundred per cent of the remaining funds that are left over after legal fees have been paid WILL be donated to charity, I refuse to shift on that stance.
"Ideally I would like to see the funds go to a children's hospital.
"Once we have had legal counsel with the expert on the ECHR I will provide further updates, but please know, this could possibly be the end of the road, but I can take some solace at least in knowing that what is left over from the money we raised can go to helping those in need.
"Thank you very much guys, I will post updates as soon as I have more information."