It was not immediately clear how many diplomats and staff would have to stop working at the main U.S. embassy in Moscow and other consulates around Russia, but sources quoted by Russian news agencies said "hundreds" of people would be affected.
Russia President Vladimir Putin, who Trump has expressed respect for, has also criticized the bill.
Senators overwhelmingly approved the bill with a vote of 98-2, a day after the House and Senate agreed on the terms. Both were veto-proof numbers.
"(The Russians) have taken Trump's measure and while they are willing to exploit his goofy fixation on Putin and naive sense you can do deals with someone like Putin. they realize his clownish performance as president makes it really hard for him to deliver on any of the big things that Russia wants", said Andrew Weiss, a former national security council Russia expert. It is aimed at punishing Moscow for interfering in the 2016 USA presidential election and for its military aggression in Ukraine and Syria, where the Kremlin has backed President Bashar Assad.
Both houses of Congress passed the bill - which also slaps sanctions on Iran and North Korea - with almost unanimous bipartisan support. Tillerson said the Trump administration needs more "flexibility".
Moscow said Friday that the United States must reduce the staff at its embassy and consulates to 450, the same number Russian Federation is allowed to have in the US. U.S. Embassy spokeswoman Maria Olson could not immediately say how many people had to leave to meet Russia's new requirements.
Before Trump's decision to sign the bill into law, Sen.
Speaking on ABC's "This Week", Russian Deputy Foreign Minister Sergei Ryabkov referred to a bill passed by Congress on Thursday to sanction his country as "weird and unacceptable", and said it was "the last straw".
Any veto attempt would have created a political firestorm for Trump given the ongoing investigations into whether members of his campaign colluded with the Russian government during the presidential campaign.
The US ambassador to Russia, John Tefft, was summoned to the foreign ministry on Friday morning to be given the news.
The North Korea sanctions are created to punish the country for its nuclear and ballistic-missile programmes.
The sanctions against Iran, modelled on previous executive orders, were created to punish entities that support terrorism, sell weapons to the country, or help its ballistic missile programme.
The bill would affect a range of Russian industries and might further hurt the Russian economy, already weakened by 2014 sanctions imposed after the Ukraine crisis.