What seems like a very complicated conversation is really much simpler than it's been made to be. Regardless of whether someone says, or even does, something objectionable or even wrong, it's no one's business, it's between them and G-d. This is the fundamental operating principle behind the prohibition of lashon hara, literally evil speech. A person is prohibited to say anything about another person, even if it is true, if it might cause people to think negatively of the person being spoken about. The Rabbis have even extended this as far as to mean that a person shouldn't even say something positive about a person, lest it cause another person to think or say something negative about the person being spoken about. A proper treatment of these laws can be found in the book Hafetz Haim, written by Rav Yisrael Meir Kagan zt"l and English books on the subject can be found from the Chofetz Chaim Foundation.
Now on a technical level once a matter is known to ten or more people, it can be spoken about in public. Nonetheless this is generally, but not always, bad policy. The fact of the matter is that we all have opinions that are going to be found to be distasteful. Actions are primary, and with very rare exceptions the only things the human beit din (courts) were put in charge of were actions. Therefore, if we know something said something disgusting in private, but does great things in public, we should focus on these deeds. That is not making light of what they said, but simply attempting to judge people favorably.
Making public racist statements is very different. The purpose of making inflammatory comments is to inflame, instigate, and divide. We see from the case of Penina, the co-wife of the then-childless Channa in 1 Samuel who teased Channa in order to get her to pray for children, but was punished for hurting her feelings by losing her own children that it is never permissible to inflict emotional harm with words. Racist comments are obviously worse since they cause to upset others with no productive value. They only cause hurt and discord.
Now some will point to some of the statements of the Rabbis in the Talmud or elsewhere that appear to be racist. This in itself is a form of demonization of Judaism. The Rabbis were astute about human nature and made insights in much the same way that psychologists and anthropologists do now, at a time when the fields were not differentiated as they are now. That distinction only really took hold with figures such as Ockham and Descartes almost 1,000 years after the Talmud was written. The Rabbis always judged individuals as individuals and to every extent they could favorably. Many examples can be brought from the Talmud and later to establish this to be the case.
First, it must be noted that racism, per se, is a hillul hashem, a desecration of God's name. This because it denies the fact that all people are created in the image of God, or, as the mishnah in Sanhedrin (4:5) writes: “For this reason the Adam was created as an individual, … so that there would be harmony among people, so that one would not say to their fellow ‘my father is greater than your father;’ and so that the sectarians should not say ‘there are many powers in Heaven.’” Racism is a notorious way of denying another human their Divine image, and by implication denying that all were created by God. This is a hillul hashem. For this reason the Babylonian Talmud categorizes the apikorus or apostate as one who “has spurned the word of the Lord.” (Numbers 15:31, Sanhedrin 99a). A hillul hashem is worse in public, but even in private desecrating the name of God is a very serious offense. According to Jewish, however, “public” is constituted by three or more people, so it is not clear to me that according to the canons of Jewish law, and even more so according to Jewish values, racist remarks made often to close friends constitute “private remarks.” Public in the Sterling case implies the media, but in fact the media just magnified the already existent desecration of the name of God.
The condemnation of a person of great power, wealth, and therefore social, cultural, and political influence because of racist remarks is a proper way for society to distance themselves from those remarks. It is an appropriate way of saying: “That is beyond the pale.”
At the same time, this condemnation, while appropriate, takes no real sacrifice on the part of the people doing the condemning. We must all work to root out racism everywhere—in our own hearts and minds first, perhaps, but also in the education system, in the justice system, in our democratic culture. We have to reform prisons and sentencing (The New Jim Crow), we have to stop criminalizing everyday life so kids (overwhelmingly black and brown kids) who misbehave are incarcerated rather than educated, we have to enforce the right to vote for everybody. That will counter the desecration of the name of God with a kiddush Hashem, a sanctification of the name of God.
It is somewhat difficult to separate out this particular issue from the individual. Mr. Sterling is a complicated man, being accused of racism on several occasions while being the recipient of a Lifetime Achievement Award by the National Association for the Advancement of Colored People award. To show the depth and history of this conundrum Sterling previously won this same award in 2009, the same year the Los Angeles Clippers general manager Elgin Baylor sued him for employment discrimination (alleging racism and ageism). So the questions you raise are important for this specific case as well as for the broader context.
Indeed you point out the crux of the matter- that a private conversation was made public. While content of this specific conversation was about racism, certain Jewish principles applicable here also cover other “topics” as well.
According to Jewish tradition there must exist a balance between private and the public realms. We learn from a medieval enactment (called a takanah), attributed to Rabbenu Gershom Me'or Hagolah, that one is forbidden from reading a letter written and sent by another person without that person's consent (found in Responsa R. Meir of Rothenburg (ed. Prague), no. 1022). Furthermore, one who violated this takanah was subject to excommunication (Shiltey Giborim to Alfasi, Shevu`ot, fol. 17a, end). Therefore we learn from this precedent that what is said or communicated in private should remain private. It is easy to extend this line of thinking to a personal phone conversation shared between friends, lovers or spouses. When an intimate exchange is taken to the worldwide media, it is easy to see the breach of privacy.
In addition, Judaism also values the principle that we should not embarrass others, so taking a private conversation public with the intent to do harm clearly violates such a tenet. Yet on the other hand there is the principle that a sinner should not be rewarded in Talmudic literature. Surely aspersions by a single (narrow minded, bigoted) individual cast upon a group of people should not be met with an increase of income of millions of dollars. I would assert that Jewish texts and values help us to see your question through a variety of lenses, and each situation must be evaluated in its own right. We learn we must strive to preserve an individual’s dignity and privacy while at the same time protecting those being persecuted, especially until there is proof of any wrongdoing.