Its decision on Monday dealt her a new blow in her struggle to secure re-election in January amid criticism of her handling of Beijing and rising tension with China.
China kick-started a campaign to isolate Taiwan and throttle its economy in 2016, when the island democracy of 23 million people elected President Tsai Ing-wen, who is bitterly opposed by the Communist government in Beijing.
Taiwan announced it was severing ties with the Solomon Islands on Monday after learning the Pacific nation had made a decision to switch diplomatic recognition to China, which it said was "extremely regrettable".
It came after months of speculation that the Solomon Islands was considering a diplomatic switch to Beijing.
"Beijing's goal is to diminish Taiwan's global presence, hurt the Taiwanese people, and gradually suppress and eliminate Taiwan's sovereignty", it said.
"The Chinese government attacked Taiwan purposely before our presidential and legislative elections, obviously aiming to meddle with the voting".
While Wu has tendered his resignation, the Presidential Office quoted Tsai as saying that his resignation is not an issue as all staff at the foreign ministry had put in their best efforts until the last minute and that the severing of ties was due to China's suppression of Taiwan.
"Taiwan has never bowed to pressure from one single setback, and it won't be defeated by this blow", Wu said, urging support from allies in the region to defend Taiwan's much-valued freedom and democracy.
Other countries which have switched diplomatic ties from Taiwan to China since 2016 include Panama, El Salvador, the Dominican Republic, Burkina Faso and Sao Tome and Principe.
The protests in Hong Kong pose the biggest challenge for Communist Party rulers in Beijing since President Xi Jinping took power in 2012.
The diplomatic switch reduces the number of countries that recognise Taiwan to 16.
The swap will enhance China's growing presence in the string of lightly populated South Pacific islands where Taiwan and the United States - which fought key battles during World War II on Guadalcanal, the Solomons' main island and site of its capital - once held sway. The prime minister's office did not immediately respond to questions.