The two met at the Oval Office on Monday morning and addressed reporters at the White House afterward.
Merkel came to Washington in part to push against arming Ukrainian forces in their conflict against Russian-backed rebels, In Brussels, EU ministers held off tightening sanctions to give peace talks a chance. Merkel is meeting with Russian President Vladimir Putin on Wednesday.
"We continue to encourage a diplomatic resolution to this issue, and as diplomatic efforts continue this week we are in absolute agreement that the 21st century cannot have us stand idle and simply allow the borders of Europe to be redrawn at the barrel of a gun," Obama said.
The German leader stressed that she did not see a way to solve the problem through military means.
Obama expressed less certainty on the subject of force.
"Both Angela and I have emphasized that the prospect for a military solution to this problem has always been low," he said. "Expecting that if Russia is determined, that Ukraine can fully rebuff a Russian army has always been unlikely."
Still, the president admitted, current actions by other Western powers have not dissuaded Russia from its actions across its neighbor's borders.
"It is true that if in fact diplomacy fails, what I've asked my team to do is to look at all options," Obama said. "The possibility of lethal defensive weapons is one of those options that's being examined. But I have not made a decision about that yet."
The president said he did not have a line in the sand for when he would send lethal weapons to Ukraine, but that a decision would be part of an effort to dissuade Russia from its current course of action.
Obama said he would continue to discuss the matter with other allies as he considers next steps, saying that inter-governmental partnerships were key to finding a resolution for the conflict.
"What I do know is that we will not be able to succeed unless we maintain the strong trans-Atlantic solidarity that has been the hallmark of our national security ... and I'm confident that I've got a great partner in Angela in maintaining that," he said.
Merkel agreed with Obama that Russia has not lived up to prior agreements about not stoking violence in Ukraine, but said more diplomacy was needed.
Speaking about the current negotiations, the German chancellor said through a translator: "I myself, actually, would not be able to live with not having made this attempt."
Still, Merkel said that even though the United States and Germany may not end up agreeing on the issue of providing defensive arms for Ukraine, that the disagreement would not strain the two nations' relations.
At a security conference in Munich over the weekend, Merkel said it was uncertain whether further negotiations would lead to a deal with Putin but argued that all opportunities for a diplomatic solution should be pursued.
Merkel, who with French President Francois Hollande is due to meet Putin on Wednesday in Belarus, has come under fire from U.S. foreign policy hawks in the Republican-controlled Congress who want defensive weapons sent to the Ukraine army.
"The Ukrainians are being slaughtered and we're sending them blankets and meals," Sen. John McCain said at the Munich conference. "Blankets don't do well against Russian tanks."
Obama will soon make a decision on whether to provide Ukraine with lethal weapons to fight the separatists, U.S. Secretary of State John Kerry said last week.
The German chancellor also addressed the European debt negotiations with Greece, saying that Germany's policy since 2010 has aimed at keeping Greece in the euro zone. Merkel said what will count "is what Greece will put on the table."
"The rules are the same, you put in efforts and on the other side you are shown solidarity," she said of Greece's commitments, adding that she awaits a "sustainable proposal" from the country.
Merkel also praised the economic recoveries from Ireland, Spain, and Portugal after a strong phase of structural reforms, potentially hinting that she hoped for the same thing from Greece.
—Reuters contributed to this report.