CNN quoted Central News Agency (CNA), a state-owned news agency operated by the Government of the Republic of China (Taiwan), reporting that Tsai apologised to the supporters of the party for the "disappointing performance" in a speech at its headquarters.
The referendums are a headache for Tsai and her ruling Democratic Progressive Party, which faces serious challenges in major seats.
Taiwan has earned a reputation as Asia's gay capital.
The poll results could also cost Tsai the 2020 election since she might not even be nominated to run, analysts said.
A referendum proposing to change the name under which Taiwan competes at global sports events that has already angered China was defeated by 5.5 million votes, with 4.6 million in favor. It was seen as a test of support for independence.
Amnesty tweeted, "Taiwan's referendum results rejecting marriage equality are a bitter blow but despite this setback, we remain confident that love and equality will ultimately prevail!"
But surveys have consistently shown that Taiwanese people prefer not to have to choose between one or the other; instead they favour the middle ground.
"She will need to express clearly that Taiwan is happy to bolster cooperation with the U.S., while she also needs to make it clear that Taiwan is not trying to lock horns with China" said Jou Yi-cheng, who was once a speechwriter for former President Chen Shui-bian, a DPP member.
China sees self-ruling Taiwan as part of its territory and is angry that Tsai will not acknowledge that the island is part of "one China", unlike her predecessor Ma Ying-jeou of the Beijing-friendly Kuomintang.
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Assange also fears extradition to the U.S. over WikiLeaks' release in 2010 of secret USA military documents and diplomatic cables. In June, WikiLeaks released emails obtained from these hacks.
Tsai stepped down Saturday night as party chief, though retaining the presidency, to take what she described as "full responsibility" for the election losses.
"This lies at the core of Taiwan's democratic values, which Beijing should respect and correctly understand", the Mainland Affairs Council said on its website. "We must study and accept the higher expectations of the people". They are anxious its pro-business and pro-unification stance could make the island too dependent on China economically and risk Beijing having too much influence over Taiwan's affairs.
While the election authority had not yet announced the final results of Saturday's referendum, more than six million voters approved a series of initiatives opposing marriage equality, passing a threshold needed for a public vote. Tsai's flagships achievements to date include pension reform, active support for renewable energy over nuclear power and stronger ties with the United States - a counterbalance to China. The Nationalists ruled Taiwan for decades after it split from mainland China in 1949 amid civil war.
Ko Wen-je, the independent candidate, won the election of Taipei mayor with about 3,200 more votes over Ting Shou-chung, a candidate from the Kuomintang.
Taiwanese President Tsai Ing-wen.
After more than three decades together, Wang Tien-ming and Ho Hsiang finally chose to tie the knot when Taiwan's top court ruled a year ago that same-sex marriage must be legalised.
The issue over Taiwan's sovereignty is a complicated unresolved remnant of history dating back more than a century and, to be fair, Chinese people have as much right to want unification as Taiwanese people do to want independence.
Tensions across the Taiwan Strait have heightened, with China conducting military drills around the island and snatching away Taiwan's dwindling number of diplomatic allies.