Gareth Lee launched his initial case against Ashers bakery in 2014, after its owners Daniel and Amy McArthur refused to write 'support gay marriage' on a cake for the customer.
In 2018 the Supreme Court ruled in favour of the McArthur family, saying they had the right to refuse to bake a cake with a slogan they didn't believe in.
Mr Lee is now appealing the decision before the European Court to refute the right of businesses to express religious beliefs.
He argues the owners' opinions should be separate from a company's.
The Christian Institute says this latest action is an attack on every person's right to freedom of expression.
The Christian Institute's Director Colin Hart said: "If it is wrong to compel an individual baker to say something they do not believe, it is just as wrong to compel a small family bakery. People don't relinquish fundamental freedoms just because they set up a family company."
The UK's highest court judges were unanimous in finding no discrimination on the grounds of religious belief, political opinion or sexual orientation in the case against the Ashers Bakery. They also ruled that equality law does not compel people to say something they profoundly disagree with and the issue was about the message and not the messenger.
Speaking in relation to the Supreme Courts ruling, The Christian Institute's Deputy Director Simon Calvert said: "They all knew of the implications to freedom of speech and religion had the decision gone against the Ashers bakery, which could have included a Muslim printer being forced to print cartoons of Mohammed, or a bakery owned by a lesbian couple being forced to make a cake describing gay marriage as an 'abomination'."
Mr Calvert continued: "I'm surprised and a little disappointed that anyone would want to overturn a ruling that protects gay business owners from being forced to promote views they don't share, just as much as it protects Christian business owners and hope the Government will robustly defend the current law."
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