"We stand together because we believe that people and nations have the right to determine their own destiny - and that includes the people of Ukraine," Obama said before a crowd of more than 6,000 that spilled into the streets before Royal Castle, a symbol of Polish independence.
"Our free nations will stand united so that further Russian provocations will only mean more isolation and costs for Russia," Obama said. "Because after investing so much blood and treasure to bring Europe together, how can we allow the dark tactics of the 20th century to define this new century?"
Obama spoke just after announcing new U.S. assistance for Ukraine's armed forces, including body armor and night vision goggles - the first such delivery of tools that would directly help Ukrainian troops in their battle against pro-Russian separatists.
Speaking from behind protective glass, Obama declared that "the days of empires and spheres of influence are over." He cast the people of Ukraine as heirs to the push for democracy in Poland that helped bring an end to communism in Europe.
"Bigger nations must not be allowed to bully the small, or impose their will at the barrel of a gun or with masked men taking over buildings," Obama said. "And the stroke of a pen can never legitimize the theft of a neighbor's land. So we will not accept Russia's occupation of Crimea or its violation of Ukraine's sovereignty."
Holding his first extended meeting earlier Wednesday with Ukraine's President-elect Petro Poroshenko, Obama praised the billionaire candy maker as a "wise selection." He said he was impressed by Poroshenko's business expertise and confident he can handle Ukraine's formidable economic and political challenges as he takes office on Saturday.
Obama offered Poroshenko $5 million in new U.S. aid as Ukraine's military continues to suffer casualties in its confrontation with pro-Russian insurgents in the country's east. More significant than the dollar amount was the nature of the new aid. Until now, the U.S. had only provided other nonlethal forms of aid like clothes, food and radios.
Obama told reporters allowed into the end of their meeting in a Marriott hotel conference room that the international community must stand with Poroshenko to make sure that Russia is no longer supporting separatists.
Poroshenko, speaking softly in English, thanked the U.S. for its support and refusal to recognize Russia's annexation of Crimea. He said the next phase is crucial to starting a peaceful process out of Ukraine's political crisis.
"We feel a friend in need is a friend indeed," Poroshenko said.
World leaders excluded Russia from a Group of 7 meeting starting Wednesday night in Brussels that was originally slated to include President Vladimir Putin and take place in Sochi, Russia. But the leaders of Britain, France and Germany have scheduled face-to-face talks with the Russian leader in the coming days, exposing divisions among Western nations that had united to isolate Russia over its aggressive moves against Ukraine.
Obama and Putin have spoken by phone multiple times - but not in person - since Russia annexed Crimea and stationed tens of thousands of troops on its border with Ukraine. Obama said Tuesday he maintains a "businesslike relationship" with Putin and is certain to encounter him in France on Friday during events to commemorate the 70th anniversary of the Allied invasion of Normandy. But they have no formal talks scheduled.
Obama foreign policy adviser Ben Rhodes said the White House believes the most important conversation is between Russia and Ukraine's new leadership. "We don't want different countries to be having conversations over the head of the government in Kiev about Ukraine's future," Rhodes told reporters aboard Air Force One en route to Brussels.
The White House is not concerned that other leaders are having talks with Putin, Rhodes said, as long as they delivering the message that Russia should reduce tensions in Ukraine and stop its support for separatists destabilizing the country. "The question is not whether there are meetings, it's what people are saying in those meetings," Rhodes said.
He said a dinner with the G-7 leaders scheduled for Wednesday night "will be an important opportunity to try to advance that alignment." He also said the leaders will discuss what Russian steps would trigger new sanctions, now that the previous warning to Putin not to interfere with Ukraine's presidential election has become moot.
Obama announced upon his arrival in Europe Tuesday that he's asking Congress for up to $1 billion to increase the U.S. military presence on the continent in the face of Russia's aggression. He reaffirmed the U.S. commitment to Poland's security under the NATO alliance - a message he repeated to the Polish people and that U.S. leaders have echoed across the region.
"These are not just words," Obama said. "They are unbreakable commitments, backed by the strongest alliance in the world and by the armed forces of the United States of America: the most powerful military in history."