Barack Obama tells Russia to respect borders of Georgia and Ukraine
Mr Obama struck a conciliatory tone for much of his wide-ranging and televised speech to students at the New Economic School, according to Telegraph.co.uk.
But he pointedly mentioned Georgia and Ukraine by name.
"State sovereignty must be a cornerstone of international order," said Mr Obama.
"Just as all states should have the right to choose their leaders, states must have the right to borders that are secure, and to their own foreign policies.
"Any system that cedes those rights will lead to anarchy. That is why this principle must apply to all nations - including Georgia and Ukraine."
Mr Obama, whose two-day visit to Moscow reflects his efforts to recalibrate US relations with a skeptical Russia, said the two countries are not "destined to be antagonists".
"The pursuit of power is no longer a zero-sum game," he said. "Progress must be shared.
"Let me be clear: America wants a strong, peaceful and prosperous Russia."
In his speech, Mr Obama said the interests of Russia and the US generally coincide in five key areas: halting the spread of nuclear weapons; confronting violent extremists; ensuring economic prosperity; advancing human rights of people; and fostering co-operation without jeopardising sovereignty.
But he also sprinkled in challenges to Russia on its own soil, particularly in the area of democracy.
"By no means is America perfect," said Mr Obama. "Independent media have exposed corruption at all levels of business and government. Competitive elections allow us to change course ... If our democracy did not advance those rights, I as a person of African ancestry wouldn`t be able to address you as an American citizen, much less a president."
Earlier, Mr Obama held private breakfast talks with the Russian prime minister Vladimir Putin at his dacha, or country home, outside Moscow. The atmosphere was cordial and both voiced hope for improved relations between Washington and Moscow.