UNICEF asked that Russia let children's "best interests" guide the "design and development of all efforts to protect children."
"We encourage the government to establish a robust national social protection plan to help strengthen Russian families. Alternatives to the institutionalization of children are essential, including permanent foster care, domestic adoption and inter-country adoption," he said.
The United States has signed but not ratified the convention, which has sparked concerns from conservatives over its impact on U.S. sovereignty and parental rights.
Groups such as Amnesty International and Human Rights Watch had urged 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, "this bill is frankly a childish response to the Magnitsky Act."