"During this uncertain period, we expect the military to ensure that the rights of all Egyptian men and women are protected, including the right to peaceful assembly, due process, and free and fair trials in civilian courts," he said.
Obama urged the new authorities to avoid arbitrary arrests and said U.S. agencies would review whether the military action would trigger sanctions on aid. A senator involved in aid decisions said the United State would cut off its financial support if the intervention was deemed a military coup.
Much may depend on a strict definition of "coup."
Sisi, head of Egypt's armed forces, stressed that the army acted to enforce the will of the people. They demonstrated in the millions against Mursi this week. Sisi said the president had failed to heed their demands.
Washington's senior general, Martin Dempsey, said that if the move by Sisi, a graduate of the U.S. Army War College, was seen as a coup it would affect relations: "There will be consequences if it is badly handled," he told CNN. "There's laws that bind us on how we deal with these kinds of situations."
Concerns over human rights have clouded U.S. relations with Cairo, but did not stop aid flowing to Mubarak, or to Mursi.
The European Union, the biggest civilian aid donor to its near neighbour, also called for a rapid return to the democratic process. Foreign policy chief Catherine Ashton said in a statement that should mean "free and fair presidential and parliamentary elections and the approval of a constitution."