The UN suspended Russia from its Human Rights Council on Thursday over atrocities in Ukraine.
Shortly after the vote, Russia's UN ambassador said it was quitting the council entirely.
Zelenskyy has since called on the UN to expel Russia from the Security Council, where it has veto power.
Russia said it was quitting the United Nations Human Rights Council on Thursday shortly after it was suspended for atrocities in Ukraine.
The UN General Assembly voted 93 to 24 to suspend Russia on Thursday, with 58 abstentions. UN regulations require a two-thirds majority of voters to expel or suspend members.
The assembly moved to vote on Russia's membership after evidence emerged of more than 300 civilian deaths during Russia's occupation of Bucha, a town outside the capital Kyiv.
Though there is substantial evidence suggesting Russia committed the killings, Russia denies the claims and says the deaths were staged to discredit Russia.
Following its suspension, Russia said it was quitting the body entirely, with Gennady Kuzmin, Russia's ambassador to the UN, saying the suspension was an "illegitimate and politically motivated step," Reuters reported.
Speaking to reporters afterwards, Ukraine's ambassador to the UN, Sergiy Kyslytsya, said: "You do not submit your resignation after you are fired," per Reuters.
Russia became only the second country to be suspended from the Human Rights Council, following Libya in 2011. The US quit the council in 2018, under President Donald Trump, but later rejoined under President Joe Biden.
Biden said Thursday that the vote was a "meaningful step by the international community further demonstrating how Putin's war has made Russia an international pariah."
The UN's suspension sends "a clear message that the suffering of victims and survivors will not be ignored," Linda Thomas-Greenfield, the US ambassador to the UN, told the UN General Assembly on Thursday, Reuters reported.
"We ensured a persistent and egregious human rights violator will not be allowed to occupy a position of leadership on human rights at the UN."
In an address delivered Friday, Ukrainian President Volodymyr Zelenskyy said Russia was planning fresh attacks on civilians around Ukraine which would be as bad as seen in Bucha.
The previous day, Zelenskyy called for Russia to be removed from the UN Security Council over the Ukraine invasion, saying the body should "dissolve" itself if it won't take the step.
As one of five permanent members, Russia has the power to veto "substantive" resolutions.
Russia's decicion to quit the council has echoes of 2014 when, following its suspension from the G8 over its annexation of Crimea, Russia quit the G8, making it the G7.