Yes, but the owner must post a sign stating their beliefs and what they refuse

Religion deserves to have no place in society because it claims a divine position to ideas invented my humans.

If the owner don’t want that stuff in their store than that stuff shouldn’t be promitited in the store.

no. religion is inherently delusional. it is not based on evidence, and therefore to deny others anything based on a religious supposition is also delusional.

No they must not because it’s no one else’s business what there beliefs are. You don’t have to have a personal relationship with them or even talk to them. Why is that what everyone thinks about they always think you can’t be friends with someone that has different beliefs but it shouldn’t matter at all.

With America being a nation built on religious freedom, I strongly believe that it people shouldn't be denied service based on their religion, even if it conflicts with the owners.

requiring a sign-posting policy legitimizes and formalizes discrimination, creating a "separate but equal" environment that violates the core intent of public accommodation laws and the 14th Amendment's Equal Protection

I don't think religious freedom applies in this regard at all. Religious freedom is solely the right to practice religion of your choice however you see fit within the social limits of the given nation.

They do not need to post their religion on a sign since that doesn't directly correlate with the value of what they are selling or the value of their own person.

everyone has a right to shop but they can refues to sell to the person if they do something agenst the law

No matter who they are, services should not be limited to certain people simply because it doesn’t align with another’s religious beliefs. Everyone reserves the right to any service unless their actions prove otherwise.

The customer isn’t forcing the owner to partake in the activity/belief-challenging ideal, and both parties can simply mind their own business. The owner wouldn’t be committing apostasy or anything for helping someone do their own thing. Additionally, segregation started as a refusal on the basis of beliefs, simply for the buyer’s identity or personal beliefs. As long as clashing beliefs aren’t being forced upon the owner, which they most likely are not, both sides can partake in their own business and treat each other as equals. It’s simply a matter of respect.

Having the right to refuse service due to a customer's beliefs opens the door to defacto segregation, where a segment of the population can face segregation and discrimination when a proponderance of business refuse to serve "their kind." It should be understood when opening a business that provides the public with goods and services that this is something that one is not allowed to do, just as it is not allowed to skip paying taxes, or abuse one's employees. If you can't step up to the responsibility of treating all people as equal, then find your livelihood somewhere else.

But if the owner's beliefs are subject to extreme political ideology and the belief hurts the economy system, would granting the owner this political power, would still be safe?

No business owners should not be able to post a sign, because they should judge someone by the way their religious beliefs are because they are still human.

No they must not because it’s no one else’s business what there beliefs are. You don’t have to have a personal relationship with them or even talk to them. Why is that what everyone thinks about they always think you can’t be friends with someone that has different beliefs but it shouldn’t matter at all.