As a rule, folks have the right not to do things they do not want to. Compelling someone to sell a cake they don't want to sell in order to sell cakes at all, well that infringes on their rights. Sure, it makes the baker seem a bit petty but they have the right to do that as well. Its hard to come up with a parallel that would apply to a liberal atheist baker on account of there always being special ways to plead the case. You could try saying, would you want to bake a NAZI wedding cake and should you be forced to in order to bake at all? Maybe, should you be compelled to bake a cake that had a specifically anti-gay marriage message for a fundies wedding?
Let me put this in a different context. Suppose you are selling something. You agree to sell it for a certain price but then have second thoughts and want backsies.
The other party does not have to agree and can legally force you to deliver.
Of course, that doesn't infringe on your rights. You bartered your rights away and doing that - contracting - is a right, too.
Now suppose you want to sell something and shout to a group: "I'm selling this for X dollars!" Someone shouts back: "Accepted."
But when that guy steps out of the group you realize that it's a guy you hate. You want backsies. The question now is whether you bound yourself when you shouted the offer or not.
In reality, either may be true depending on various details. It's a technical matter. No one would seriously argue that one possible solution infringes on anyone's rights.
So let's look at the baker. If he is forced to make the cake, then that means that he bound himself by offering his service to the public. If that were the legal status quo, it would not seem like an obvious infringement on anyone's rights.
But let's try yet another way of looking at it. Suppose the baker does not own his bakery but is employed by a chain. That guy is going to get in trouble if he refuses paying customers. Does that infringe on his rights?
Actually, I do see the point here. Being self-employed is nice. It's nice being your own boss. It's freedom, right?
So maybe we really should do something about this. Maybe change the tax code to make big chains like Starbucks or Pizza Hut impossible in favor of independent, single establishments? Whatever the case, not being able to refuse a customer is certainly a tiny thing in comparison.
What if nazi wanted a cake? Let's up it a notch. A nazi pedophile who raped the baker and his kids comes in a and wants a cake. Can the baker's employer force him to serve that customer?
IDK but there are certainly things that an employee can't be made to do. There are situations where you could get out of a contract based on exceptional hardship. I think here we are close to the issue. Are gay weddings so terrible that a person should not have to acknowledge their existence?
That's a question of societal values; not of rights.
I can think of a rights issue in regards to this. Suppose it's the only bakery in town. Getting a cake from out of town would cost extra. In that case the baker has a position of economic power and can effectively levy a fine on gay wedding cakes.