MOSCOW—President Vladimir Putin formally ordered a suspension of Russia’s obligations under a Cold War-era nuclear treaty with the U.S., his office said Monday, a death knell for the pact that heightens the threat of a new arms race.

The Kremlin said the decision to suspend the agreement was made after the U.S. said it would stop abiding by the pact, the Intermediate-range Nuclear Forces Treaty, which prohibited the possession, production and flight testing of certain short and intermediate-range nuclear missiles.

The...