Lets say you own a Flower Shop and you really like Tulips... you like Tulips better than Roses. You want to put the Tulips in the front of the store and keep the Roses in the back. It's your store- you should be able to do that, no?
Yes. As long as no local laws are contravened.
The Google issue is somewhat different in that they provide a service in a different country where the laws might not be the same as they are in the USA.
Here's a clear example:
In New Zealand, if someone commits a minor crime that falls within particular parameters (I think where the offence carries a maximum jail sentence of 3 months or less) then that crime, or crimes, can not be divulged by the Police or anyone else after a period of 2? years have passed. Even if a potential employer makes inquiries of the Police they can't divulge those crimes. It is illegal to do so.
Enter Google ...
Google or any other search engine would be breaking local law if they made those particular crimes available to the general public. It isn't unlawful but illegal to do so. The law expressly forbids it.
If I had a minor criminal record that met the criteria where those crimes were forbidden by law to be divulged, and I noticed that the service provided to New Zealand by Google identified me as committing those crimes, then I could take measures to force Google, or any other search engine of my choosing, to remove that information being provided by the service that they offer this country. I could do this because Google would be breaking local law. Google, or any other international/multinational company, that provides services to New Zealand do so knowing full well that they must comply with our laws.
Does that mean that Google is the only search engine committing that crime? No, of course not. But New Zealand law only extends to our borders. Our government can't prevent someone, say from America, hosting that information on their website but it can prevent that information being disseminated to the New Zealand public. Google allows that dissemination.