In fact, protesters began calling Morsy "Morsilini," a reference to the late Italian fascist dictator Benito Mussolini who was Adolf Hitler's ally. That nickname arose after Morsy gave himself sweeping powers in November.

Morsy later canceled most of those powers following demonstrations. That turn of events hurt Morsy's image because he was enjoying international attention for playing a constructive role in the recent, bloody conflict in Gaza between Hamas and Israeli forces, analysts said.

The stakes are high for a country strategically positioned in Middle Eastern politics and in world trade through the Suez Canal.

"I don't think the international community can afford for (Egypt) to collapse economically ... or politically," Cook said.

The defense minister's warning is "very important" because "it shows the military has been in consultation about this. That's why I take it more seriously," Cook added.

In the coming month, Egyptians will go to the polls to elect a lower house in Parliament. The election will be a bellwether on how Morsy's Muslim Brotherhood now stands against the opposition coalition National Salvation Front, analysts said.

"They are smart people," Stacher said of opposition leaders, "but the problem is that they don't seem like they want to have a real democracy either."

For now, the Egyptian military doesn't appear to want to intervene and run the Egyptian government again as another president is selected.

"If the situation deteriorates further, the military might not have a choice and it might find a warm reception," Cook wrote on his blog for the Council on Foreign Relations.

In a revolution, the first government typically doesn't stay in power, as seen in the Russian and French revolutions, Coyle explained.

"Usually it gets replaced by more radical elements of society," he said.