He isn't. Not legally, nor should he be in a society where . . .
Yes he should be. If he's a Democrat and does not want to sell a cake to a Republican Party event, he should be free to refuse, if they tell him this is what it's for. If it's for a Nazi Party event, he should be free to refuse. But if they don't tell him, he has to sell them the product. What is wrong with this rule?
. . . where people are interdependent and rely on the exchange of goods and services, and goods and services providers are dependent on the communities and infrastructures which support them.
The providers will be more abundant and be able to serve consumers better if they are free to use their discretion. But if they are forced to violate their conscience or serve a cause distasteful to them, there will be fewer providers to serve consumers.
The alternative is that vendors must serve any customer even if they are told by the customer that the product is for a use the vendor disagrees with.
By that rule, the vendor would have to serve the Neo-Nazi group throwing a Hitler birthday party. Such a party is not illegal.
If you say NO!, the vendor could refuse to serve the Neo-Nazi group, what is the rule? That Neo-Nazi party is perfectly legal. Just because the Nazis did crimes in the 1930s does not mean it's illegal today for someone to be a Nazi and hold a party for Hitler.
If you make an exception for Nazis, then what about other offensive persons in history? What about a Jefferson Davis birthday party? what about a Karl Marx party? Lenin? Christopher Columbus? Attila the Hun? Genghis Khan? Dracula?
What is the rule? There are many evil causes someone might want to celebrate and want to order a cake for. Is the vendor required to serve EVERY imaginable customer, no matter how obnoxious, as long as the activity or use of the product in question is legal? Then you would require that Jewish baker to provide a cake to the Neo-Nazi event.
How about a birthday cake for Mao Tse Tung? Mao murdered many more millions of humans than Hitler did. So, should an anti-communist baker be required to provide a birthday cake for a Mao Tse-Tung birthday party?
I get it. You think gays and lesbians are mass murderers. Dictators who establish horrible regimes and kill millions.
How about answering the question: Should that baker be required to serve the group throwing a party for Mao? Yes or no?
This does not equate Maoists with gay weddings, except that both are legal, and there is no rule saying why a vendor must serve one but may refuse to serve the other. Until you give that rule,
YOU are the one equating them.
The best rule is that it should be the vendor's discretion, if the customer makes the mistake of saying what the product is to be used for.
There has to be a stated rule. You can't make up rules as you go along, judging each case only on emotional impulse.