An example scenario: Alice is a victim of a crime, reports the crime and Bob is arrested and goes on trial. Bob pleads not guilty and Alice participates in the trial as a witness. Bob is sentenced, the court record is made. The Daily News (fictional paper) reports on the court records of the day and has a reporter who attends the more interesting cases, and mentions Bob's sentence and gives some of Alice's statements as quotes.
In that scenario, the court record should always be a matter of public record, a statement of fact. The newspaper certainly has the right to access public record and to make a news story of the set of facts that are in the public record.
But, here starts the problems... Alice applies for a job and the employer Googles her name and comes across the news article. There are many types of crimes in which the public have great difficulty accepting a victim is a victim. For example, rape. It isn't too much of a stretch to say that the culture of victim blaming means that a matter of public record has just had the effect of defaming Alice.
Alice as a victim is never given the opportunity to move on with her life when every person that ever searches for her will find the story very quickly. She has been sentenced too by participating in the justice system, which is an open book.
The newspaper, just as in this case, will argue this is public record and cannot be silenced. Sure, I agree... but that doesn't mean that it's in the victims interest that the information be extraordinarily easy to find.
And Google are a better place in which to attempt to stop the information being found, given that they (and only 1 or 2 other search engines) cover the vast majority of searches made about someone.
Alice certainly does have the right to make information that she didn't explicitly choose to make public and that can cause her harm not be found so easily, even when that information is a matter of fact and public record.
She has the right to not be found (by that method - Google).
PS: I know a girl experiencing almost exactly that scenario, who cannot get a news story off of the front page results for her name. This isn't even a stretch scenario. The local newspaper just hasn't bothered responding to requests.