So pedantic...Public would be anything which represent the public interest such as elected officials, public rallies, petitions, etc.
You included the term to be defined in your definition. "The public would be...public"?
In any case, whereas I disagree with some of the reasoning of St. Thomas Aquinas, the following articles are worth noting:
"I answer that, A law, properly speaking, regards first and foremost the order to the common good. Now to order anything to the common good, belongs either to the whole people, or to someone who is the viceregent of the whole people. And therefore the making of a law belongs either to the whole people or to a public personage who has care of the whole people: since in all other matters the directing of anything to the end concerns him to whom the end belongs" (ST I-II, q. 90, a. 3, corpus).
"I answer that, Whatever is for an end should be proportionate to that end. Now the end of law is the common good; because, as Isidore says (Etym. v, 21) that "law should be framed, not for any private benefit, but for the common good of all the citizens." Hence human laws should be proportionate to the common good. Now the common good comprises many things. Wherefore law should take account of many things, as to persons, as to matters, and as to times. Because the community of the state is composed of many persons; and its good is procured by many actions; nor is it established to endure for only a short time, but to last for all time by the citizens succeeding one another, as Augustine says (De Civ. Dei ii, 21; xxii, 6). "((ST I-II, q. 96, a. 1, corpus).
Operation of a business would include opening, operating, engaging with the public in trades of goods and services for money (or other goods and services, i.e., bartering).
I disagree with this definition. A business rarely engages with the public qua public. A business generally deals with private individuals.
So, the public (government, rally, etc.) doesn't have the right to discriminate against businesses but the businesses (should) have the right to discriminate against the public. That is a public entity should not bar a business from operation based on arbitrary reasoning.
You've misunderstood me. The public (that is, the law) does not have the right to discriminate against individuals, since the law concerns itself with the common good of the entire people. The individual does not have the right to discriminate against the public (that is, the State). That said, the State (the public personified), since it deals with law governing the whole people and not with individual matters...
But this or that black person (or even all of them) are not the public. They are private individuals.
Where is the line then? We can conclude that racism, sexism, etc. are moral matters as they negatively affect a portion of society, similar to prostitution. So why can't the government step in and say "you are not allowed to discriminate arbitrarily" versus the government saying "you are not allowed to sell your body."
The government has the right to say what kind of exchanges are acceptable and which are not. The government generally does not seem to have the right to compel this individual to engage in this particular exchange. The right of the individual to engage in this legally acceptable exchange rightly falls under the law as permissive, not as compulsive. [For permissive vs. compulsive, see ST I-II, q. 92, a. 2.]