It's all in that headline.
Unless there's criminality involved, there is no legal means to compel changes to the search results. But if someone is infringing your trademarks, or helping others access copyrighted material, you can easily get them removed, because by ranking pages that promote those illegal activities, the search engine is complicit. For instance, let's say you have an online casino usa oriented website whose unique content has been completely copied by another online casino affiliate website. And to make matters worse, their online casino website is now ranking above your site in a google search. They obviously are infringing on your copyrighted content. It's one thing to use information from another similar casino website and rephrase it. Or to use information from a number of online gambling sites, rewriting it to make the content unique. However a word of word rip off of content is not acceptable. Report this online casino site to Google immediately.
So if you can prove illegality, you can influence the search - actually get websites removed. But as far as contesting your site's ranks in Google legally - unless you know a way around a First Amendment defense, forget it. The search results have been ruled to be an opinion, therefore protected speech. It's been tested many times and is now a matter of legal precedent.
The same rules apply if you don't like seeing unflattering content ranking for your business or personal name. Unless it's slanderous, it stays. And if it's slanderous, it's not the search engine who has to deal with it. But if the sites are held offshore by interests not under US law, you may find it impossible to identify a target for legal action. This means that legal recourse to bad publicity even when addressing criminal behavior, may not be possible.
Even withing US law, legal action is risky. Here, the metric for acceptability is truth, the telling of which may be brought under contention, legally. But by doing so, you expose yourself to this liability: many courts are hesitant to shut down websites that do not clearly violate ethical standards because of First Amendment concerns. As lawyers quickly learn, there's no slam-dunk legal way to influence what people see online.
A more productive approach is to treat the problem as an SEO issue regarding web pages carrying good stories about the enterprise & the individuals involved in it. Good press can usually be made to outrank bad, and this permits the telling of the company story on its own terms - something that may have lasting value by itself. Ranking positive pr high on page 1 for all trademark and principal keywords is good for business and also insulates you from weak sites ranking with bad press.
The conclusion regarding legal action to change search results is not favorable. Especially when court action is compared to a pr campaign deploying SEO on positive stories, legal strategies end up being expensive and risky. An SEO solution involving good pr has permanent value, carries no risk of legal blowback, and is very likely to be less costly.
Read more about higher level enterprise seo.