Officials in Moscow vented their frustration with the punitive action, warning that it will erode global stability and fuel conflicts.
In an emotional Facebook post, Russian Prime Minister Dmitry Medvedev described the move as a humiliating defeat for Mr Trump.
The Russian Foreign Ministry has warned of possible new retaliatory measures.
"The hope for improving our relations with the new US administration is now over," Mr Medvedev, who served as Russian president in 2008-2012 before stepping down to allow Vladimir Putin to reclaim the job, said.
The Kremlin had been encouraged by the Trump's campaign promises to improve the Russia-US ties that had grown increasingly strained under President Barack Obama.
But with the White House preoccupied by congressional and FBI investigations into links between the Trump campaign and Russia, the hoped-for relationship reset has not materialised.
"Trump's administration has demonstrated total impotence by surrendering its executive authority to Congress in the most humiliating way," Mr Medvedev said.
"The American establishment has won an overwhelming victory over Trump," he added. “The President wasn't happy with the new sanctions, but he had to sign the bill. The topic of new sanctions was yet another way to put Trump in his place."
Mr Medvedev emphasised that the stiff new sanctions amount to the declaration of an "all-out trade war against Russia".
"We will continue to work calmly to develop our economy and social sphere, deal with import substitution and solve important government tasks counting primarily on ourselves," he said. "We have learned how to do it over the past few years."
Asked on Wednesday whether Moscow planned additional steps in response to Mr Trump signing the bill, Mr Putin spokesman Dmitry Peskov answered that "retaliatory measures already have been taken”.
But shortly after, the Foreign Ministry warned that "we naturally reserve the right for other countermeasures”.
It said the sanctions bill reflects a "short-sighted and dangerous" attempt to cast Russia as an enemy and would erode global stability. The ministry added that "no threats or attempts to pressure Russia will force it to change its course or give up its national interests”.
The ministry said: "We are open for cooperation with the US in the spheres where we see it useful for ourselves and international security, including the settlement of regional conflicts".
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