The United States is concerned about President Mohamed Morsy's decision to assume sweeping powers, the US State Department said on Friday.
Morsy on Thursday issued a decree that puts his decisions above legal challenge until a new parliament is elected, causing angry protests by his opponents and violent clashes in central Cairo and other cities on Friday.
The European Union called on Morsy to respect the democratic process.
"It is of utmost importance that (the) democratic process be completed in accordance with the commitments undertaken by the Egyptian leadership," a spokesman for EU foreign affairs chief Catherine Ashton said in a statement.
These commitments included "the separation of powers, the independence of justice, the protection of fundamental freedoms and the holding of democratic parliamentary elections as soon as possible," he said.
Morsy's aides said the decree was intended to speed up a protracted transition that has been hindered by legal obstacles, but rivals condemned Morsy as an autocratic "pharaoh" who wanted to impose his Islamist vision on Egypt.
US Secretary of State Hillary Clinton met Morsy in Cairo on Wednesday and thanked him for his mediation efforts to establish a ceasefire between Israel and Hamas, ruling the Gaza Strip.
"The decisions and declarations announced on November 22 raise concerns for many Egyptians and for the international community," State Department spokeswoman Victoria Nuland said in a statement.
"The current constitutional vacuum in Egypt can only be resolved by the adoption of a constitution that includes checks and balances, and respects fundamental freedoms, individual rights, and the rule of law consistent with Egypt's international commitments.
"We call for calm and encourage all parties to work together and call for all Egyptians to resolve their differences over these important issues peacefully and through democratic dialogue."
Egyptian police on Friday fired teargas near Cairo's Tahrir Square, the heart of the 2011 uprising that toppled former President Hosni Mubarak. Thousands demanded Morsy quit and accused him of launching a "coup." There were also violent protests in Alexandria, Port Said and Suez.
Mubarak was an ally of the United States for decades. His downfall has thrown into doubt the United States' long-standing reliance on Egypt, the first Arab state to make peace with Israel as a strategic partner in the region.
Clinton said on Wednesday: "Egypt's new government is assuming the responsibility and leadership that has long made this country a cornerstone for regional stability and peace."
