Ebru Umar, a well-known atheist and feminist journalist of Turkish origin, said she was arrested at her home in Kusadasi, a resort town in western Turkey, early on Sunday and brought before a judge.
She later said she was "free but forbidden to leave the country".
Umar, 45, recently wrote a piece critical of Erdogan for the Dutch daily Metro, extracts of which she then tweeted.
Dutch blog Geenstijl said they had received an sms message from Umar saying that she had been arrested after someone reported her tweets on a hotline set up by the Turkish officials.
The hashtag #freeebru was trending in the Netherlands Sunday, with Dutch politicians and commentators calling for her release.
Dutch Prime Minister Mark Rutte said in a tweet that he "had had contact with @umarebru last night. Our embassy is in close contact with her for assistance".
Education Minister Jet Bussemaker told Dutch WNL television "it is absurd that you can be arrested for a tweet."
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The foreign ministry told AFP it was following the case closely and was in contact with authorities in Turkey.
Umar, who reportedly became a journalist under the influence of Theo van Gogh -- a Dutch filmmaker later murdered for making a controversial film about Islamic culture -- had written about a diplomatic spat between Turkey and the Netherlands.
A political storm erupted this week over reports an email sent by the Turkish consulate to Turkish organisations in the Netherlands asked people to forward any emails and social media posts which insult Erdogan or Turkey.
Rutte said he would ask Ankara to clarify the call. The Turkish consulate said the note was sent by a consular official who used an "unfortunate choice of words" that was misinterpreted.
The case followed outrage in Germany after the government there gave a green light for authorities to begin criminal proceedings against popular comic Jan Boehmermann for performing a satirical poem about Erdogan.
Trials in Turkey for insulting Erdogan have multiplied since his election to the presidency in August 2014, with nearly 2,000 such cases currently open.
In September 2015 freelance Dutch journalist Frederike Geerdink was deported from Turkey after being detained during clashes between Kurdistan Workers Party (PKK) rebels and Turkish security forces.
It was the second time she had been taken into custody: in April Geerdink was put on trial on charges of spreading "terrorist propaganda" for the PKK but was then acquitted.