Legally, you can certainly ignore this request. In some circumstances, it's possible that hosting content that is illegal in Europe this would make it illegal to provide access to your site, which could even lead to a court decision requiring ISPs to block access to your site. I don't know if it's even legally possible for that to happen for a denied right-to-be-forgotten request (I think it's happened for sites hosting child pornography), and anyway the effective likelihood for things to go that far is nil. Given that the complaint is from one disgruntled user, even if you were based in Europe, it's unlikely that they'd go far enough to get a court judgement against you.
A second point is that it isn't clear from your description and from my understanding of the law whether it applies in this case. The court ruling that sparked this (it's a court ruling, not a law — it's a consequence of European directives, mainly Directive 95/46/EC Article 12) applies to automated search engine, but human-curated content would be treated similarly: the point of the ruling is that linking to content about an individual bears similar responsibilities to hosting that content. Note that the ruling did not conclude the existence of a right to be forgotten — there's a discussion about this in the EU, but no law as yet. The ruling judged that the information was “inadequate, irrelevant or no longer relevant, or excessive in relation to the purposes of the processing at issue carried out by the operator of the search engine” (§94–99), in violation of Directive 95/46/EC Article 6. It was instrumental in the ruling that searching for the person's name brought up a disproportionate amount of links about the 1998 affair. Posts on an obscure forum would probably fall under the same category, since they'd have to be dug up.
I think that under current law, these forum posts wouldn't be affected, but they might be if the proposed right-to-be-forgotten legislation enters the books. Of course, this is not legal advice, this the uninformed opinion of some guy on the Internet who just spend 5 minutes reading Wikipedia.
If the links are to news articles which are incorrect, they might fall under libel law, which is a different kettle of fish altogether.
Beyond the legal issue, a smear campaign is rarely pretty. Whether the guy is genuinely an asshole or not, it's probably better for the well-being of the forum to cut it down. If nothing else, it sets a bad climate, showing to newcomers who've never heard of the guy that it's ok to make personal attacks on people. My advice would be to let it go quietly. If the posting of news articles was a one-time thing, either delete the thread or let it remain forgotten somewhere, depending on whether deletion is something you normally do. If it's recurrent, put a stop to it, gently, telling people they surely have something better to do thatn discuss a user who's left months after he's left.
