Speaking Saturday at the Munich Security Conference in Germany, Medvedev said he sometimes found himself wondering whether this was 2016 or 1962.
"NATO's policy with regard to Russia has remained unfriendly and opaque. One could go as far as to say that we have slid back to a new Cold War," Medvedev said. "Almost on an everyday basis we are called one of the most terrible threats either to NATO as a whole or to Europe, or to the United States."
Tensions between the West and Russia have increased in recent years, in large part -- at least in the view of the West -- due to Russia's annexation of the Ukrainian peninsula of Crimea and its support for separatists elsewhere in eastern Ukraine.
More recently, some in the West have questioned whether Russia's intervention in Syria is helpful. Russia says it is attacking terrorists. But some observers contend that Moscow is intent primarily on propping of the regime of Syrian President Bashar al-Assad, who is hanging onto power despite a five-year civil war.
Gen. Philip Breedlove, NATO's supreme allied commander Europe, told CNN that NATO does not agree with Medvedev's assessment. At an earlier briefing at the Munich Security Conference, Breedlove said Russia is not just trying to change the rules but rewrite them.
"We at NATO do not want to see a Cold War," he said in an interview. "We do not talk about it. It's not what we want to happen or anticipate to happen... We're a defensive alliance who are arraying ourselves to face a challenge ... [from] a nation that has once again decided it will use force to change internationally recognized borders and so we take those appropriate actions to be able to assure, defend and deter."
The back and forth came as Secretary of State John Kerry told the Munich conference that Russia's attacks in Syria have been largely "against legitimate opposition groups" and that must change.
The Syrian war has raged for five years, destroying once-great cities, killing nearly a half million people, and setting several million to flight in a historic migratory wave that Kerry acknowledged was fraying the social fabric of Europe.
Kerry and Russian Foreign Minister Sergey Lavrov met briefly at the conference to discuss plans for a cessation of hostilities in Syria, the State Department said in a statement.
They also discussed the establishment of a United Nations task force to coordinate humanitarian aid, according to a communique issued by the International Syria Support Group.
Kerry and Lavrov agreed on the need for that aid to begin flowing as rapidly as possible, State Department spokesman John Kirby said in a statement.
Referring to the conflict in the Ukraine, Kerry said earlier that Russia's choice in the matter was simple: Either fully implement the Minsk agreement or face economically damaging sanctions.
"Russia can prove by its actions that it will respect Ukraine's sovereignty just as it insists on respect for its own by the same token," Kerry said, with Lavrov in the audience.
The secretary of state announced that the U.S. will significantly upgrade its commitment to European security, with a planned "four-fold increase in our spending on the European Reassurance Initiative," from just under $790 million to $3.4 billion.
"This will allow us to maintain a division's worth of equipment in Europe and an additional combat brigade in Central and Eastern Europe, making our support and NATO's more visible and more tangible," he said.
World powers, including the United States and Russia, this week agreed to a ceasefire in Syria and to the delivery of immediate aid there.
In Syria, the Russian military has stepped up its presence by land, air and sea, and Russian officials have contended their weaponry is targeting ISIS extremists and their infrastructure.
But some analysts have likened the Syrian conflict to an emerging proxy war between Russia and the United States, harkening back to the Cold War.
U.S. officials have accused the Kremlin of using its military to support al-Assad, an ally, and targeting anti-regime rebels.
Pressure has been mounting on Russia to work with the international community
in determining which groups in Syria to attack -- and which, instead, deserve a seat at talks on a peaceful future for the country.
The Cold War pitted East against West and pushed the world to the brink of nuclear war. The struggle between communism and capitalism defined the second half of the 20th century. The tension began after World War II and ended with the collapse of the Soviet Union in 1989.
The NATO military alliance was formed after World War II by countries in North America and Western Europe. It has 28 member states committed to defending each other.
Russian President Vladimir Putin has been accused of trying to undermine the unity of NATO, particularly with the destabilization of Ukraine and the annexation of Crimea. Putin has announced that will add more nuclear missiles and build a new generation of non-nuclear ones that could strike U.S. soil.