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But he told reporters that the attack on the American consulate in Benghazi would not break the bonds between the US and the new Libyan government.
Mitt Romney, Mr Obama's political foe, criticised the handling of the crisis.
Ambassador J Christopher Stevens reportedly died of smoke inhalation after a crowd stormed the consulate.
Three other Americans were also killed, amid reports that rocket-propelled grenades had been fired during the assault on Tuesday night.
The consulate was set ablaze and a charred car could be seen parked near the damaged buildings on Wednesday.
A US marine anti-terrorism team is being sent to Libya to bolster security after the attack, a US defence source told reporters in Washington.
Protesters against the film also attacked the US embassy in Cairo on Tuesday night.
Speaking in the Rose Garden at the White House, President Obama told reporters: "Make no mistake. Justice will be done."
He said he condemned "in the strongest possible terms the outrageous and shocking" attack.
The killing of the US ambassador to Libya, Christopher Stevens, along with three other US embassy staff, raises a host of questions about security, the prevalence of armed militias in the country and the whole trajectory of Libya's democratic project.
The Obama administration in Washington will want answers from the Libyan government about the scale of the security measures in place at its consulate in Benghazi and how demonstrators were able to get into the building.
More broadly the ambassador's death highlights the continuing instability in Libya as the country struggles to establish security and the rule of law.
The country is awash with weapons and armed militias - some of a salafist or extreme Islamic fundamentalist outlook - hold sway in many areas.
"It is especially tragic that Chris Stevens died in Benghazi because it is a city that he helped to save," he added, praising the dead ambassador for his work in Libya after the overthrow of the late Col Muammar Gaddafi.
Libya's interim leader, Mohammed Magarief, apologised to the US over the killings, which he called "cowardly criminal acts".
Reports say a militia known as the Ansar al-Sharia brigade was involved in the attack, but the group has denied the claim, the BBC's Rana Jawad in Tripoli says.
Ambassador Stevens and his staff went to the consulate in an attempt to evacuate the site after it was attacked, the Associated Press news agency said.
The building apparently came under attack by a crowd armed with guns and rocket-propelled grenades.
"The American ambassador and three staff members were killed when gunmen fired rockets in their direction," a Libyan official in Benghazi told Reuters news agency.
The Libyan doctor who treated Mr Stevens in hospital said he had died of severe asphyxiation, apparently from smoke inhalation, with no other injuries, and that he had tried for 90 minutes to revive him.
Time correspondent Ashraf Khalil describes the scene on the ground in Egypt and concludes the reaction to the film was "essentially a case of an American group of fringe Christian fundamentalists successfully provoking and enraging a similar group of fringe Muslim fundamentalists".
ABC's Jake Tapper explains the chronology of events in Egypt and Libya as the Romney campaign accused the White House on Tuesday of being sympathetic with those who waged the attacks.
NBC calls Mr Romney's criticism of the Obama administration "one of the most over-the-top and incorrect attacks of the general-election campaign".
In the International Herald Tribune, Harvey Morris shares Christopher Stevens' recent emails with him about his hopes for Libya.
He was the only American brought into the Benghazi Medical Centre and initially nobody realised he was the ambassador, Ziad Abu Zeid told AP.
Mr Romney, Mr Obama's Republican challenger in this November's presidential election, said the Obama administration had appeared to "sympathise with those who waged the attacks" in Benghazi and Cairo.
Mr Obama's team, he said, had sent "mixed signals to the world" in the face of violence.
Our correspondent says many people in Libya are still armed following the conflict that overthrew Gaddafi.
Analysts say the attack will raise serious new concerns about stability in the country and the ability of the new Libyan administration to maintain security.
In June, two British bodyguards were injured in an attack in Benghazi on a convoy carrying the British ambassador to Libya. Red Cross and UN staff also came under attack this year.
Correspondents say the film at the heart of the row, which appeared on Youtube translated into Arabic, is highly provocative and insulting to Muslims.
A demonstration in Cairo on Tuesday saw protesters breach the US embassy and tear down the US flag, which was flying at half-mast to mark the 9/11 attacks.
An Islamic tenet bans the portrayal of the Prophet Muhammad.
Cartoons featuring the founder of Islam sparked violent unrest among Muslims in 2005 when they were published by a Danish newspaper.
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