Ashers Bakery in Belfast was told to pay £500 last week after it refused to bake the cake for the man on the basis it went against its Christian beliefs.
In a statement, the McArthur family said: "After much careful and prayerful consideration given to legal advice, we have decided to appeal the judgement handed down last Tuesday.
"We continue to insist that we have done nothing wrong as we have discriminated against no individual but rather acted according to what the Bible teaches regarding marriage.
"As many other people have already noted, Christian beliefs seem to have been trampled over in this judgement and we believe this only has negative effects for our society."
Daniel McArthur, his wife Amy & two children
The family added: "Our hope and prayer would be that an appeal will allow us and other Christians to live out their faith in Jesus Christ in every part of their lives, including their workplace."
The case was brought against the bakery by Northern Ireland's Equality Commission.
The bakery is being supported by The Christian Institute. Simon Calvert, from the organisation, said: "I believe that most people think that this is a ruling that should be overturned.
"There has been such extraordinary support from people from all walks of life who are appalled by what has happened to the McArthur family.
"There is huge public support for an appeal and it is vitally important that the higher courts consider this issue."
He added: "This court decision will have dramatic consequences if it goes unchallenged."
Northern Ireland's Equality Commission told Premier it had not yet received any communication about an appeal but would give it "consideration" when it does.