Jack Phillips from Masterpiece Cakeshop was told in 2014 by the Colorado Civil Rights Commission that it was unlawful to refuse the work and that he should re-educate his staff.
Two years earlier he had declined an order claiming his faith meant he could not design a cake promoting same sex marriage.
With the backing of religious freedom charity Alliance Defending Freedom, lawyers for Mr Phillips say the government cannot coerce Phillips to create artistic expression that communicates a message with which he fundamentally disagrees.
They're calling on justices to afirm his constitutional right to refuse the order.
"Nobody should be forced to choose between their profession and their faith," said Kristen Waggoner, Senior Counsel for ADF.
"Phillips gladly serves anyone who walks into his store, but, as is customary practice for many artists, he declines opportunities to design for a variety of events and messages that conflict with his deeply held beliefs.
"In this case, Jack told the couple suing him he'd sell them anything in the store but just couldn't design a custom cake celebrating their wedding because of his Christian faith."
The case mirrors the legal action faced by Ashers Bakery in Norther Ireland.
The owners were found to have discriminated against a customer based on his sexual orientation after they refused to make a cake which said 'support gay marriage'.
Having been told to pay compensation, they're still trying to overturn the decision.