"While welcoming Prime Minister Dmitri Medvedev's call for the improvement of the child welfare system, UNICEF urges that the current plight of the many Russian children in institutions receives priority attention," he said.
He asked that Russia let children's "best interests" guide the "design and development of all efforts to protect children."
Groups such as Amnesty International and Human Rights Watch had urged Russian lawmakers to reject the bill.
"This bill hits back at Russia's most vulnerable children and could deprive them of the loving families they desperately need," Hugh Williamson, Europe and Central Asia director at Human Rights Watch, said last week.
John Dalhuisen, Amnesty International's Europe and Central Asia program director, has said that "this bill is frankly a childish response to the Magnitsky Act."