I'm still trying to get clarity regarding whether the Objectivist ethics regards privacy as a right. Is gaining unauthorized access to a website containing personal information an initiation of force? Is releasing this information to the public an initiation of force? Would either action constitute a crime in a society and legal system based on Objectivist principles?

The entire basis for the right is its superiority to any authority and it exists for every person from birth prior to any contract. This is the basis of individual sovereignty which every American has from birth. It describes a right that is understood to be inalienable which is the reason that government cannot infringe upon it, and that the right exists for all people. The purpose of the statement in the amendment is to restrict government but the point is that the right comes before government even exists. Therefore that right exists for every person and can not be infringed by any other entities unless consent is given to do so.
Consent is the crux of the issue. You give consent and you lose the right and any protection that the constitution may afford. It becomes a contractual matter and that is the case with the websites. For example, Google can do anything they wish with your data if you agree to the vague terms of their service agreement.

I think the 4th Amendment mostly relates to unreasonable searches and seizures by the government. How does the Objectivist ethics view a public disclosure of personal information by another private individual? Would it be considered an initiation of force?

I would consider it to be a right described in the 4th amendment, but in most commercial situations it results from the agreement (consent) by the customer (user of the web site) to the site's contractual terms. Consent is given, although it may come from a naive person, ignorant of the value that they are agreeing to sacrifice.
(The 4th amendment does not create the right; it describes a right that all sovereign people have from inception unless they consent to give it up.)

Is there an Objectivist view as to whether privacy is a right? In her book For The New Intellectual Ayn Rand said, "Civilization is the progress toward a society of privacy." But I haven't seen any mention of whether she regarded privacy itself as a right. Does the Objectivist ethics regard public disclosure of personal information as an initiation of force?

Its the young people who don't understand or value privacy (yet) that are a target market imo.

You know it's like these cockroaches getting popped with kiddie porn. They should burn them down not so much for porn or Ashley Madison or whatever. They should burn them down because stupidity at that level is mind boggling.