The constitution does not just restrict the government, it does occasionally also specifically restricts people. For example, it prevent certain people from becoming President of the USA. But that is mostly a side point.
The main thing you seem to misunderstood is that the US Constitution restricts what the government can do, but also ALLOWS the government to do certain things. Things like shutting down a business that refuses to serve black people, Jews, or gay people, etc.
But most importantly, you are mis-stating the problem. What is going on here is a bunch of prejudiced people are insisting they have certain rights when they do NOT have that rights. Religious rights are the rights to worship as you wish, they do NOT have the right to do engage in business as they wish. You can't go around claiming that your religion gives you the right to chop people's heads off.
In this particular case, the reach is EXTREME. The religion they claim to worship has no clear and concise ruling for homosexuality. It certainly does NOT mention selling anything to gays, let alone wedding related gear. The religious duty they claim is unwritten, just a matter of personal beliefs that is NOT shared by the majority of people that have that religion. They have not issued a re-written bible, clearly stating their claim. If you won't re-write your holy book to add your so called religious duty, then either you are not serious about the claim or you are not serious about your religion. On top of that, no such claim among Protestants was made until AFTER the law was passed (although one of the ancient roman Christian Emperors did outlaw gay marriages and murdered existing couples). It's not something protestants have been doing since Martin nailed his notice to the door, nor do modern Catholics allow it. Their claim for 'religious' is highly suspect.
But most importantly, this so called religious right violates laws that were not designed or written to oppress the religious people. It is legal to outlaw beheading and cocaine and you can't use "religious right" to over-ride that law. You can only use 'religious right" to over-ride a law that does not make reasonable exceptions.
The reasonable exception for a gay wedding is to sell the cake, but refuse to do an inscription and refuse to sell a cake topper with two men/women.
These idiots don't want to do that, they refuse to be reasonable. They want the right to refuse to sell a cake that looks like a wedding cake to someone that is gay because they MIGHT possibly use it as a wedding cake.