The decision will be seen as a victory for EU regulators, who are ambitious to see US tech giants meet tightened European standards over hate speech and offensive content.
Last week, the same court decided that Google was not legally compelled to apply the EU's strict "right to be forgotten" rules globally, in a victory for the search giant.
In a closely watched judgment, the European Court of Justice said EU law "does not preclude" courts from ordering "the removal of information or to block access worldwide," a statement said.
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The latest case was brought originally to an Austrian court by Greens party politician Eva Glawischnig-Piesczek, who requested the removal of Facebook posts that the judges found defamed her and could be seen by users of the social network around the world.
The complaint also concerned messages from fictitious accounts, which according to the Greens, had called Glawischnig-Piesczek a "corrupt" person and which the social network refused to delete.
A higher Austrian court referred the case to the EU's top court for an opinion and the judgment, which cannot be appealed, will now be used as a reference Europe-wide.
With the decision, Facebook and similar platforms such as Twitter, face a greater obligation to monitor their content and take down content found to be offensive or hateful, even from fake accounts.
Facebook slammed the EU court's decision, saying "it undermines the long-standing principle that one country does not have the right to impose its laws on speech on another country."
It also deplored the obligation to track down "equivalent" content that duplicates offensives or hateful language.
"In order to get this right, national courts will have to set out very clear definitions on what 'identical' and 'equivalent' means in practice," a Facebook statement said.
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"We hope the courts take a proportionate and measured approach, to avoid having a chilling effect on freedom of expression," it said.
Glawischnig-Piesczek, the victim of the hate speech, hailed the decision as "a historic success for human rights against web giants".
"It in no way infringes freedom of opinion," Glawischnig-Piesczek told the Austrian news agency APA.
EU-wide rules on hate speech are limited.
So far, online giants including Google's YouTube, have agreed to voluntarily take down hateful or dangerous content, including those linked to terrorism, within 24 hours.
However, the EU is expected to propose tougher Europe-wide measures including fines if Facebook and others fail to comply with orders.
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