Biden on Monday rejected the idea that his comment could escalate tensions over the war in Ukraine or that it would fuel Russian propaganda about Western aggression.
"Nobody believes ... I was talking about taking down Putin," Biden said, adding that "the last thing I want to do is engage in a land war or a nuclear war with Russia".
He said he was expressing an "aspiration" rather than a goal of American foreign policy.
"People like this shouldn't be ruling countries. But they do," he said. "The fact they do doesn't mean I can't express my outrage about it."
Biden's remark in Warsaw ricocheted around the globe despite the White House's swift attempts to clarify that the president only meant that Putin "cannot be allowed to exercise power over his neighbours or the region".
On Monday, United Nations Secretary General Antonio Guterres responded to Biden's speech by saying that "we need de-escalation. We need military de-escalation and rhetoric de-escalation".
Although Biden has frequently touted American unity with European allies since the invasion of Ukraine began, he appears to have caused some discomfort by targeting Putin in Warsaw.
French President Emanuel Macron said Sunday he "wouldn't use those terms, because I continue to speak to President Putin, because what do we want to do collectively? We want to stop the war that Russia launched in Ukraine, without waging war and without escalation."
Secretary of State Antony Blinken was forced to continue clarifying Biden's speech during a trip through the Middle East, where he had intended to focus on solidifying American partnerships as the administration seeks a renewed nuclear agreement with Iran.
Speaking at a news conference in Jerusalem, Blinken said Biden meant that "Putin cannot be empowered to wage war or engage in aggression against Ukraine or anyone else."
Biden has previously gone further than expected when speaking about Putin, describing him as a "war criminal" at a time when administration officials were saying they were still conducting a review of the matter.
White House press secretary Jen Psaki said then that Biden was "speaking from the heart" rather than articulating a legal conclusion.
Republicans questioned why Biden decided to go off-script in Warsaw when dealing with a combustible conflict.
Some said his provocative rhetoric was strange given his otherwise cautious approach, such as refusing to facilitate the transfer of Polish fighter jets to Ukraine's military.
"If we're so worried about provoking him that we couldn't even send MiGs into Ukraine, how is this any different?" Rep. Michael McCaul, R-Texas, told CNN's State of the Union on Sunday. "In fact, I would say it's more provocative than sending MiGs into Ukraine."
The US has been rushing weapons like anti-tank missiles into Ukraine, and is considering providing anti-ship missiles to make it harder for Russia to mount an amphibious offensive along the Black Sea coast.
Ukrainian President Volodymyr Zelenskyy remains exasperated with the pace of military assistance, accusing Western leaders of cowardice and repeating his request for tanks and fighter jets.
Associated Press writer Edith Lederer contributed to this report from the United Nations.
