That private property owners have the ability to restrict the rights of others on their property is not without precedent. In my non-lawyer-ish opinion, it's lawful and legal for them to do so.
That is true, within limits. To limit the ability of guests, patrons, or customers on your property to speak, publish, or worship while on your property are more or less permissible.
Even here however, employment laws have extended protections to employees whose religious beliefs require the wearing of certain clothing such as the Jewish yarmukle against adverse employment actions in most cases. Whether that is viewed as a proper exercise of government to protect minority rights, or an over-reach of power that denies private property owners their rights most often depends on a person's personal political views, and the particulars of the case at hand.
However, where there is far less debate is whether a business or other private property owner can engage in conduct that creates a hazard to employee, customer, or patron health and safety. I don't know that I have a "right" to some minimum amount of personal space that any court would recognize as being enforceable against a private business. But we've determined that commercial venues have maximum occupancy allowances based on such factors as how many emergency exits are present and long it takes to clear a room in an emergency. This is based not on a right to personal space, but on a social value of protecting life over property or control of property.
As we saw in a recent discussion here, even most gun owners would limit the use of deadly force to defense of life and limb. Most would not allow deadly force to be used in defense of mere property.
Break into my cabin to vandalize or steal and I will expect both restitution and punishment for your conduct. Break into my cabin to avoid freezing and/or starving to death while lost in the woods, and I'd expect you to pay for or help repair any damage, but would refuse to press charges even if some prosecutor actually thought the judge wouldn't laugh him out of court in such a case. Life trumps mere property.
Our laws impose a host of health and safety regulations that could be fairly deemed "infringements" of property owners' rights to control their property or run their businesses exactly as they see fit. From building codes including seismic standards, fire sprinklers and smoke detectors, and emergency exits, to OSHA workplace safety laws (which admittedly often go too far in my opinion), and child labor laws, as a society/government we've long since decided that life trumps property.
So the question is, are we talking about mere "gun rights", or are we talking about protecting life and limb?
Now, I freely admit, this is a stretch legally because of social and legal views toward RKBA. Notice that the other side asserts that the presence of privately carried firearms creates a hazard rather than mitigating one. There are also deeply (and sometimes not so deeply) held religious beliefs centered around weapons. I don't know that Jews would claim that the presence of a firearm would desecrate or otherwise be offensive to a Synagogue, but if I'm not mistaken, it is generally considered a violation of Sabbath observance for a Jew to carry a weapon on that Holy day. The LDS Church has made clear its desire not to have (non-LDS-security) privately carried firearms in its houses of worship in Utah. I can't find any doctrinal basis for that, but neither is there any LDS doctrine against candles, but as a matter of reducing fire risk the LDS Church prohibits lighted candles in its buildings even as many other denominations routinely incorporate candles in their worship services.
Personally, I support the right of churches and home owners (or renters) to ban firearms or most any other item they don't care to admit to their houses of worship and homes. I am less sympathetic to businesses generally open to the public...especially so long as current anti-discrimination laws are in place. In a perfectly Libertarian world, I might not be the first person to suggest requiring businesses to allow the carry of guns contrary to the desires of the business owner. But in our current legal and political environment, I don't think those who legally and peacefully (and most often discreetly) carry a personal firearm for self-defense should be the only group without legal protections to go about their daily life without facing needless barriers.
And when it comes to businesses, it isn't a matter of me having a "right" to carry my gun on their property. It is about defense of my life and limb being more important than their views on guns (or fire exits, how many people they'd like to cram into a theater or dance hall, or not wanting to accommodate wheelchairs).
Life and limb trump property.