They ruled that Attorney General John Larkin QC's bid came too late.
Mr Larkin sought to refer the verdict against Ashers bakery to the UK's highest court under devolution issues.
In October, the Appeal Court upheld a ruling against the bakery's Christian owners that their refusal to make a cake iced with the slogan "Support Gay Marriage" was discriminatory.
A long-running legal battle ensued when customer Gareth Lee sued after his order was declined at the company's Belfast city centre shop in May 2014.
The gay rights activist had requested a cake depicting Sesame Street characters Bert and Ernie with the slogan for an event to mark International Day Against Homophobia.
Bakery bosses refunded his money for the order because the message went against their Christian faith.
The McArthur family, who run the firm, insist their problem was with the cake and not the customer but Mr Lee claimed he was left feeling like a lesser person.
Last week, lawyers for the bakery were formally refused leave to appeal to the Supreme Court, though the McArthur family can now petition directly for a hearing in London.
Mr Larkin backed the family's case but his attempt to invoke his powers has now been rejected.
He was intending to ask the Supreme Court to rule on issues related to devolution.
Lord Chief Justice Sir Declan Morgan ruled in the Court of Appeal that this step came after the case had ended.
"We do not consider there are exceptional circumstances in this case which require us to re-open proceedings," he said.

Speaking on TV, Maria Zakharova said Jews had told her they donated both to Mr Trump and Hillary Clinton.
She joked that American Jews were the best guide to US politics.
The diplomat's remarks caused shock. Anti-US propagandists in the last century peddled an idea that rich New York Jews controlled US politics.
Ms Zakharova was speaking on a chat show on Russian state TV at the weekend but her comments drew more attention after being picked up by media outlets on Thursday.
She said she had visited New York with an official Russian delegation at the time of the last UN General Assembly, in September.
"I have a lot of friends and acquaintances there, of course I was interested to find out: how are the elections going, what are the American people's expectations?" she said.
"If you want to know what will happen in America, who do you need to talk to? You have to talk to the Jews, of course. It goes without saying."
At this, the TV studio audience applauded loudly.
"I went here and there among them, to chat," she continued.
Imitating a Jewish accent, Mrs Zakharova said Jewish people had told her: "'Marochka, understand this - we'll donate to Clinton, of course. But we'll give the Republicans twice that amount.' Enough said! That settled it for me - the picture was clear.
"If you want to know the future, don't read the mainstream newspapers - our people in Brighton [Beach] will tell you everything."
She was referring to a district of Brooklyn with a large diaspora of Jewish emigres from the former Soviet Union.
Russian opposition activist Roman Dobrokhotov wrote on Twitter (in Russian) that the spokeswoman had "explained Trump's victory as a Jewish conspiracy".
Michael McFaul, the former US ambassador to Moscow, commented on Facebook, "Wow. And this is the woman who criticizes me for not being diplomatic."
During the election campaign, Mrs Clinton accused Mr Trump of posting a "blatantly anti-Semitic" tweet after he used an image resembling the Star of David and stacks of money.
Mr Trump, whose son-in-law Jared Kushner is Jewish, dismissed the accusation as "ridiculous".
An exit poll by US non-profit J Street suggests an overwhelming majority of US Jews voted for Hillary Clinton in the presidential election.
Northern Ireland's top law officer cannot refer the "gay cake" case to the Supreme Court, senior judges have said.
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A spokeswoman for Russia's foreign ministry has said Jewish people in New York told her they had mainly backed Donald Trump in the US election.