Obama's rival in the US presidential race, Mitt Romney, called for a tougher line with Egypt after protesters scaled the compound wall and tore down the US flag on Sept. 11.
Police clashed with demonstrators for four days after that incident and barriers were erected to stop them getting near the compound.
In a letter, Obama repeated Washington's condemnation of the film and said he looked forward to working with President Mohamed Mursi to build on the "strategic partnership", Mursi's official Facebook page said on Sunday.
"In his letter, President Obama thanked the Egyptian president for Egyptian efforts to secure the mission of the United States in Cairo," according to the site.
Egypt was a close ally of the United States under Hosni Mubarak, whose 30-year rule was ended by a popular uprising last year. The US government, a major aid donor to Egypt and long wary of Islamists, only opened formal contacts last year with the Muslim Brotherhood, the group that propelled Mursi to power.
Obama told a Spanish-language network this month that the United States did not consider Egypt's Islamist government either an ally or an enemy.
When asked whether he thought of the United States as an ally in an interview with the New York Times published on Sunday, Mursi said: "That depends on your definition of ally."
The newspaper also said Mursi indicated Egypt would not be hostile to the West but would not be as compliant as Mubarak.
"Successive American administrations essentially purchased with American taxpayer money the dislike, if not the hatred, of the peoples of the region," he said, referring - according to the newspaper - to backing dictatorial governments over popular opposition and supporting Israel over the Palestinians.
Speaking before travelling to New York for the UN General Assembly, Mursi said Washington needed to change its approach to the Arab world, show greater respect for its values and help build a Palestinian state to reduce pent up anger in the region, the newspaper said.
"As long as peace and justice are not fulfilled for the Palestinians, then the treaty remains unfulfilled," he said, referring to Egypt's US-brokered peace treaty with Israel that was signed in 1979.
Mursi, who travels to New York on Monday, is not scheduled to meet Obama during his US trip.
A spokesman for Mursi could not immediately be reached for comment, nor could a US embassy official.