The vote in the lower house of parliament was overwhelming, The New York Times reported. The measure must still pass the upper house next week and be signed by President Vladimir Putin, who has not yet said whether he will sign it.
U.S. Ambassador Michael McFaul said Russian children now in institutions would be the losers.
"If it becomes law, the legislation passed today will needlessly remove the path to families for hundreds of Russian children each year," McFaul said in a statement. "The welfare of children is simply too important to be linked to other issues in our bilateral relationship."
Incidents in which Russian adoptees were mistreated in the United States sparked the legislation. It was named after Dmitri Yakovlev, an adopted toddler who died in 2008 after being left in a hot car in Virginia.
The adoption ban was a response to the Magnitsky Act, a U.S. law named after Serge Magnitsky, a human rights lawyer who died in police custody in 2009. The law imposes economic sanctions on Russian officials found to have been involved in human rights abuses.
Lawmakers expanded the Russian bill to include all countries that approve sanctions on Russia, RIA Novosti reported.