I hope the SCOTUS rules in favor of the website designer.
In the first place, it is a private business, and no one is entitled to the labor or services of anyone absent an agreed upon contract. If someone does not wish to work for you, how can you compel them without creating an issue of servitude?
There already exists contract law that can punish someone financially for violation of a contract.
There already exists criminal law that can punish someone for fraud, embezzlement, and other financial crimes of contract violation.
But there should be no law that compels someone to work for another, because that is bondage, servitude, slavery.
Meanwhile, why not seek other "open-minded" providers of such services? There are many people who would have no problem doing the job.
So what is the REAL issue here that would compel someone to serve another?
That is indeed, the question at hand.
I can see cases for when it could happen - if a group is being effectively denied
access to an entire sector and the state lacks a
less coercive means of pursuing it's interests in ensuring access to that sector:
Example: a gay woman requires access to medication to keep a life-threatening condition at bay; all the medical providers in the area refuse to serve her because being gay is icky, the state has an interest in saying "someone is going to provide life-sustaining medication to this woman".
That fits roughly with the test established by the Religious Freedom Restoration Act that Bill Clinton signed into law: that the government may only
substantially burden the free exercise of religion of a person or organization if the government 1) has a
compelling interest to do so, and 2) is using the
least restrictive means possible to further that compelling interest, applying
strict scrutiny when adjudicating these types of cases.
...which agreeably would be vanishingly rare, especially nowadays. But if you had to draw out what would allow us to compel one person to serve another, that's where I would look.