Von Storch's Twitter account was blocked for several hours on Monday over a suspected breach of rules on hate speech.
Ms Von Storch's tweet - which she subsequently posted on Facebook as well - was apparently a reference to 2015 New Year's Eve celebrations in Cologne, which were marred by mass sex assaults on women by men of mostly migrant backgrounds.
"What the hell is going on with this country?"
Beatrix von Storch, deputy leader of the party's parliamentary faction, had criticised Cologne police for sending a New Year's greeting in Arabic on Twitter. Von Storch, whose grandfather served as finance minister under Hitler, clarified that her computer mouse had slipped.
"Do they think they will calm the barbaric, Muslim, group-raping hordes of men this way?"
Cologne police told AFP its complaint alleges that the tweet flouted laws against incitement to hate.
On her return to the Twitter, Von Storch posted in German: "Facebook has now also censored me".
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Cologne Chief Prosecutor Ulf Willuhn said prosecutors must now decide whether Ms von Storch's message amounted to "inciting people to commit violence against a certain section of society".
Ms von Storch declined to comment when asked in a newspaper interview about the police complaint, but she added it was important to speak freely about problems in Germany without censorship and the threat of charges.
AfD leaders called the actions censorship, saying the German authorities were acting like the Stasi in communist East Germany.
Under newly introduced German hate laws, social media platforms must respond quickly to remove hate speech or face fines of up to 50 million euros ($60m).
Mr Gauland encouraged users of such websites to "keep publishing" the erased comments from the AfD members.
Alternative for Germany seized 92 parliamentary seats in the September elections-the strongest showing for a far-right party in the post-war era-as it capitalized on discontent over the more than one million asylum seekers who have arrived in Germany since 2015.