There are a ton of these data aggregators; new ones pop up all the time and you can't expunge all your public records, so the short answer is maybe not entirely. But you might be able to put a nice dent in the amount that is easily available online.
The good news is that many of the data collectors have ways to opt out and have your data removed from their sites by making a request online, by fax or via physical mail. Some require things like your e-mail address for verification purposes or a copy of your driver's license or other state ID to prove your identity, although having to send them more personal information is disquieting. You can usually find opt-out instructions on their sites, if they exist, but there are places that gather lists of data brokers and their opt-out methods, if available, including UnlistMy.Info and the Privacy Rights Clearinghouse.
The bad news is some of the data might re-aggregate on the sites you had expunged, or pop up on other similar sites. Some people-finder sites may also have multiple listings for you due to name variations, misspellings and other issues, so you may have to do multiple searches and put in multiple requests.
In the event that you've submitted an opt-out request to a data broker and it reposts or fails to remove your data, you can lodge a complaint with the Federal Trade Commission (FTC). And you can contact other governmental or political groups to help lobby for policy changes, such as making some of the more egregious practices illegal.
If asking a large number of sites to remove your data seems like a Herculean task, and you don't want to wait for an act of Congress, there are growing numbers of sites that say they can delete your data from many of the data aggregators. Some of these providers include Reputation.com, Abine and MyID.com. Prices range from low monthly fees to hundreds of dollars, depending upon the company and level of service you require. Many of them have automated methods of finding and removing your data from the popular people search sites. Some will also undertake manual methods for requesting removal if they can't be achieved online, or advise you how to do it yourself, and alert you when your data pops up online again.
You might even find personal data on a regular Web site that isn't a data aggregator, and there likely won't be a formal removal process, but you may be able to contact the site owner and ask politely that your data be removed. Google itself has a form you can step through to request removal of content either from its search results or other Google properties, like Google Plus, YouTube or Picasa. There are no guarantees, but it never hurts to try.