"We think provocative rhetoric like this regarding nuclear weapons is dangerous adds to the risk of miscalculation, should be avoided, and will not indulge in it," White House Press Secretary Jen Psaki said at a briefing.
The Russian invasion - the biggest assault on a European state since World War Two - has failed to achieve the decisive early gains that Putin would have hoped for.
The United States imposed new sanctions on Russia's central bank and other sources of wealth on Monday, and many Western companies began to disentangle themselves from their Russian operations.
Over the weekend, some Russian banks were barred from the SWIFT international payments system.
Reuters
The rouble plunged 32% against the dollar before recouping about half of its losses, and Russia's central bank cranked up its key interest rate to 20% from 9.5%. Authorities told export-focused companies to be ready to sell foreign currency.
The invasion has brought relations between the United States and Russia, the world's two biggest nuclear powers, to their worst point in years.
The United States expelled 12 Russian diplomats at the United Nations, citing national security concerns. Russia described the move as "hostile".