Protester: "If it had been a white boy who had been murdered, and a black guy who had murdered him, would they have acquitted him?"
Mr Obama said Mr Martin's death was a tragedy for America, but that it was "a nation of laws and a jury has spoken".
The case sparked a fierce debate in the US about racial profiling.
The Department of Justice says it is investigating whether a civil case can now be brought against Mr Zimmerman.
Mr Zimmerman, 29, was cleared of all charges in relation to Trayvon Martin's death at the trial in Sanford, Florida on Saturday.
Prosecutors had argued that Mr Zimmerman shot Trayvon Martin dead on 26 February 2012 because he had racially profiled him as he walked through his neighbourhood wearing a hooded sweatshirt.
Trayvon Martin was African-American. Mr Zimmerman, who was carrying out area patrols after a spate of break-in, identifies himself as Hispanic.
The defence said he had killed Trayvon Martin in self-defence after the teenager punched their client, slammed his head into the pavement and reached for Mr Zimmerman's gun.
'Jury has spoken'
On Sunday, Trayvon Martin was remembered in many church services across the country. Protests also continued, with those taking part saying justice had not been done.
Trayvon Martin's relatives say they are "hurt" and "disappointed" over the verdict
In his statement, Mr Obama said the death of Trayvon Martin "was a tragedy. Not just for his family, or for any one community, but for America."
He acknowledged the case had elicited "strong passions", but said: "We are a nation of laws, and a jury has spoken."
He said all Americans should respect the call for calm reflection from the Martin family and should reflect on how to prevent future tragedies.
"We should ask ourselves if we're doing all we can to stem the tide of gun violence that claims too many lives across this country on a daily basis... As citizens, that's a job for all of us.
"That's the way to honour Trayvon Martin."
Mr Obama had commented on the Zimmerman case in March last year, saying: "If I had a son, he'd look like Trayvon."
On Sunday, the Department of Justice said it was evaluating the evidence to see whether Mr Zimmerman could face prosecution under federal criminal civil rights statutes, and whether such action would be "appropriate in accordance with the Department's policy governing successive federal prosecution following a state trial".
Civil rights groups in the US have also called for calm, though have expressed their dismay at the verdict.
The department of justice must intervene to take this to another level." He said "the American legal system has once again failed justice". But he also appealed for calm, saying anyone seeking to "compound our pain with street justice" would do "damage to the innocent blood and legacy of Trayvon Martin". Rights activist Al Sharpton said the verdict was "a slap in the face to the American people".
He compared the case to the beating of African-American man Rodney King by police in 1991, which sparked widespread rioting.
A petition launched by the National Association for the Advancement of Colored People (NAACP), demanding that the department of justice open a civil rights case against Mr Zimmerman, had received more than 350,000 signatures by midday on Sunday. Protest marches were staged in US cities including San Francisco, Philadelphia, Chicago, Washington and Atlanta. In Oakland, California, some protesters started small fires and smashed windows. Mr Zimmerman's family and representatives have said they are afraid he could fall victim to revenge attacks. His brother, Robert said he had received frequent threats on social media and there was "more reason now than ever to think that people are trying to kill him".
"He's going to be looking over his shoulder the rest of his life," he said. His lawyer, Mark O'Mara, told ABC News on Sunday that Mr Zimmerman had no regrets about having carried a gun on the night of 26 February 2012.
He said his client would now be entitled to have his weapon returned to him and to carry it, saying there was "ever more reason now, isn't there? There are a lot of people out there who actually hate him, though they shouldn't." Jorge Rodriguez, a friend of Mr Zimmerman's, said he had told him of his relief. He told Reuters he did not understand the anger at the verdict.
"Everybody asked for justice, and they got it. Everybody asked for George to be arrested, and they got it. Everybody asked for George to be tried, and they got it. Everybody asked for a fair trial, and they got it."
"It was nothing about racism. It was about the community being robbed and broken into, and one man stood up," he said. "The state should be giving this man an award, and instead they took him to trial."
BBC News
Today, Ghana remembers its year-old loss. At 14.05 GMT last year, the then President, now Late President John Evans Atta-Mills passed away. Ghanaians look back into his life and miss him. Throughout the country, people have been given the chance to express themselves as to what they want to be done in his memory. 
