Putin hasn't said whether he will sign the measure into law if it passes its next stage of being approved by the upper house.
Some top government officials including the foreign minister and the education minister have spoken flatly against the bill, one part of a larger measure by angry lawmakers retaliating against a recently signed U.S. law that calls for sanctions against Russians deemed to be human rights violators.
It nonetheless received strong approval in Friday's third reading in the State Duma, passing by a vote of 420-7-1. The upper house, the Federation Council, is likely to consider the measure on Wednesday, vice-speaker Alexander Torshin was quoted as saying by the Interfax news agency.
Torshin said there is "serious basis for supposing the draft bill will be supported by the Federation Council."
Originally the bill was more or less a tit-for-tat response, providing for travel sanctions and the seizure of financial assets in Russia of Americans determined to have violated the rights of Russians.
But it was expanded to include the adoption measure and call for the banning of any organizations that are engaged in political activities if they receive funding from U.S. citizens or are determined to be a threat to Russia's interests. In addition, it calls for anyone with dual Russian-U.S. citizenship to be banned as members of political organizations.
The U.S. said the adoption law would needlessly stop hundreds of Russian children from finding families.
"The welfare of children is simply too important to be linked to other issues in our bilateral relationship," U.S. Ambassador Michael McFaul said in a statement.
The bill is a dramatic demonstration of two strains of animosity toward the United States. The Russian political establishment resents the United States for allegedly meddling in the country's internal affairs; Putin has charged that opposition protests over the past year were the work of U.S.-funded troublemakers. Many Russians are angered by cases of adopted children abused in America and by the alleged lenience of courts in these cases.
The Duma bill is named in honor of Dima Yakovlev, a Russian toddler who was adopted by Americans and then died in 2008 after his father left him in a car in broiling heat for hours. The father was found not guilty of involuntary manslaughter.
Anger over abuse peaked in 2010 when an American woman sent her 7-year-old adopted Russian son back to Moscow on a plane alone, saying he had emotional problems and she could no longer care for him.
Despite abuse cases, Russian critics of the bill say it would ultimately victimize orphans by depriving them of an opportunity to escape often-dismal Russian orphanages. There are about 740,000 children without parental custody in Russia, according to UNICEF. Russians historically have been less inclined to adopt children than in many other cultures.
More than 60,000 Russian children have been adopted in the United States in the past 20 years, McFaul said.
But Russia's children's ombudsman Pavel Astakhov, one of the strongest critics of U.S. abuse cases, says the solution is for Russia to adopt a national program to improve orphans' prospects.
"It's necessary to strictly hold to the principle of priority for Russian adopters," he told Interfax after the Duma vote.
(Copyright 2012 by The Associated Press. All Rights Reserved.)