MOSCOW—Russian President Vladimir Putin challenged U.S. leadership in world affairs on Wednesday, arguing that an era of U.S. hegemony has come to an end as he touted Moscow’s growing military strength and increasingly assertive foreign policy.

Mr. Putin spoke during an annual event at which he answers questions from ordinary Russians. The remarks come two weeks after a summit with President Biden in Geneva where the two leaders sought to ease tensions but achieved little tangible progress. At the same time, Mr. Biden depicted the Kremlin leader as presiding over a country increasingly isolated by sanctions and struggling economically.

At Wednesday’s event, Mr. Putin pushed back, portraying the U.S. as a waning power.

“The world is changing dramatically,” he said. “On the one hand, our partners in the United States understand this, therefore there was a meeting in Geneva. On the other hand, they are trying at all costs to maintain their monopoly position.”

At their meeting earlier this month, Messrs. Biden and Putin acknowledged that relations between the U.S. and Russia have reached a post-Cold War low but both expressed hopes that the summit would allow for greater cooperation over time.