The White House welcomed the vote and denied that Obama has not spoken out as strongly as the House of Representatives.
Robert Gibbs, the White House spokesman, said he believed "the language in the reslution is very consistent with the language the president used".
Obama has voiced concerns over the protests but warned that it was "not productive" to be seen as "meddling" in Iranian politics.
Mike Pence, the Republican congressman who co-sponsored the resolution, said he believed the US should place more pressure on the Iranian government.
"When Ronald Reagan went before the Brandenburg Gate, he did not say 'Mr. (Mikhail) Gorbachev, that wall is none of our business,'" said Pence, of President Reagan's famous exhortation to the Soviet leader to "tear down that wall".
Howard Berman, the Democrat chairman of the House Foreign Affairs Committee and co-sponsor of the resolution, said "it is not for us to decide who should run Iran, much less determine the real winner of the June 12 election".
"But we must reaffirm our strong belief that the Iranian people have a fundamental right to express their views about the future of their country freely and without intimidation."
Ron Paul, a Texas libertarian and former presidential candidate who often speaks out against what he regards as government meddling, cast the sole opposing vote.
Opposition figures, led by Ahmadinejad's main challenger Mir Hossein Mousavi, have said that the vote was rigged and hundreds of thousands of Iranians have held daily street protests since the poll results were announced.
Amnesty International, the UK-based human rights group, said on Friday that it believed 15 people had been killed as the protests spilled over into violence, compared with just seven deaths reported on Iranian state radio.