At stake as Obama and his aides wrestle with that question in the coming days is the $1.5 billion in aid the United States sends to Cairo each year - almost all of it for the military - as well as the president's views on how best to promote Arab democracy.
If the United States formally declares Morsi's ouster a coup, US law mandates that most aid for its longtime ally must stop. And that could weaken the Egyptian military, one of the country's most stable institutions with long-standing ties to US authorities.
Further complicating Obama's calculus is the fact that millions of Egyptians rallied in favor of Morsi's departure, and that the military announced a roadmap for return to civilian rule that was blessed by Egypt's Muslim and Christian religious leaders.
But Morsi and his Muslim Brotherhood retain backing from a broad swath of Egyptian society, even as he alienated many of his countrymen.
Obama, after meeting top advisors at the White House, said in a statement that he was "deeply concerned" by the army's actions and had directed the relevant US agencies to review the implications for US assistance to Egypt.
But he did not use the word "coup" and stopped well short of advocating for Morsi's reinstatement, suggesting Washington might be willing to accept the military's move as a way to end a political crisis in a nation of 83 million people struggling with severe economic difficulties.
Recent history suggests Obama might take his time on deciding the future of US aid to Egypt, and by extension, Washington's relations with the country.
US law bars "any assistance to the government of any country whose duly elected head of government is deposed by military coup d'etat or decree."
When Honduran President Manuel Zelaya was ousted in June 2009, Washington temporarily suspended aid, but did not cut about $30 million in assistance until more than two months later.
Even then, Secretary of State Hillary Clinton did not determine as a matter of law that a coup had taken place.
Eric Trager, an expert on Egyptian politics at the Washington Institute for Near East Policy, said Obama should not label Morsi's ouster a coup, nor cut off US aid.
"The Obama administration should recognize that as undemocratic as a coup is, it was the result of the basic fact that President Morsi had completely lost control of the Egyptian state," Trager said by telephone from Egypt.
"Democracy was not the primary thing at stake in Egypt these last few months," but rather Morsi's mismanagement and fears of collapse of the Egyptian state, he said.
In announcing Morsi had been deposed and the Egyptian constitution suspended, Egypt's army commander promised a quick political transition. The military laid out plans for elections and a constitutional review.
Calls for quick transition
The US chairman of the joint chiefs of staff, Gen. Martin Dempsey, warned Egyptian military leaders of consequences if Morsi's overthrow were viewed as a coup.
"At the end of the day it's their country and they will find their way, but there will be consequences if it is badly handled," Dempsey told CNN.
Obama also warned against further violence, indicating that Washington's ultimate decision on aid to Egypt will depend on how Egypt's armed forces handle the transition in coming weeks.
Morsi, in power for a year, was widely blamed for presiding over a steady decline in Egypt's economy and for failing to form a broad-based government that included other groups that had joined in the 2011 revolution that overthrew President Hosni Mubarak.
US lawmakers also welcomed Morsi's departure but called for a quick transition back to democratic rule - with a close look at the aid budget.
Democratic Senator Patrick Leahy, who heads the Senate subcommittee that oversees foreign aid, on Wednesday promised a review of the $1.3 billion in military aid and $250 million in economic aid sent to Cairo each year.
Washington has cut off aid following military coups several times before. In April 2012, the United States suspended at least $13 million of its $140 million in annual aid to Mali following a coup in the West African nation.
Programs that did not go directly through government ministries were not affected.
Any Obama support for Egypt's new government is unlikely to face opposition from Republicans in Congress, who had been skeptical of Morsi's Islamist government.
"Morsi was an obstacle to the constitutional democracy most Egyptians wanted," said Republican Representative Ed Royce, the chairman of the House of Representatives Foreign Affairs Committee.
Republicans also voiced strong support for Egypt's military, whose close ties to Washington stretch back to the 1979 Israeli-Egypt peace accords.
"The Egyptian military has long been a key partner of the United States and a stabilizing force in the region, and is perhaps the only trusted national institution in Egypt today," said US Representative Eric Cantor, the No. 2 Republican in the House.
"Democracy is about more than elections," he said.