The 32-year-old was arrested on Thursday in the northeastern town of Alexandroupoli, where the couple live with their two young children.
Police searched their home after the woman filed a complaint of domestic violence and alleged he supported IS.
Officers found data on his phone that they said support the allegations of IS links, a police official told AFP.
The couple arrived in Greece in June 2016, landing on the island of Leros following the peak of the migrant crisis.
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