US Secretary of State John Kerry said meanwhile that Iran had contributed to the collapse of the government's authority in Yemen, where a Shiite militia has seized the capital Sanaa.
The French woman, who was working for an international organisation in Sanaa, was abducted on today morning, France's foreign ministry said, urging all French citizens "to leave the country as fast as possible".
Western nations including Britain, France and the United States closed their embassies in Yemen this month and urged their citizens to leave over security concerns.
A Yemeni security source said gunmen seized the woman and her local guide while they were riding in a taxi in the centre of the capital.
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Yemen has descended into chaos since the militia, known as Huthis, swept into Sanaa from their mountainous northern stronghold last year.
The Huthis overran Sanaa in September and installed a "presidential council" this month after Hadi and Prime Minister Khalid Bahah tendered their resignations.
Hadi retracted his resignation today after escaping house arrest in Sanaa, staking a claim to lead the country out of crisis.
An aide to Hadi said the 69-year-old had sent a letter withdrawing the resignation to parliament, which had never met to formally accept it.
Hadi urged lawmakers to cooperate with him "to normalise the security and economic situation in all provinces" in the letter, a copy of which was seen by AFP.