Morsy gave a speech this week highlighting his achievements during his first year in office.
But opposition members said he did not address his shortcomings and called for snap presidential polls, a new government and constitutional amendments.
The Muslim Brotherhood, whose leaders included Morsy before his election, has lost four members to violence in recent days.
Two people were shot dead Thursday when armed men attacked Muslim Brotherhood offices in Zagazig, Morsy's hometown, spokesman Gihad Haddad said. The gunmen were shouting "Down with Morsy" during the assault, he said.
Amnesty International called on authorities to uphold Egyptians' right to peaceful assembly and protect protesters and bystanders from violence.
"Given the appalling track record in policing demonstrations, it is absolutely imperative that the Egyptian authorities issue very clear instructions to security forces to uphold protesters' right to freedom of assembly and refrain from unnecessary or excessive force," said Hassiba Hadj Sahraoui, Amnesty International's Middle East and North Africa deputy director.
"They should make clear that anyone responsible for arbitrary and abusive force will be brought to justice."
In the past year, about 80 people have been killed during protests and other political violence largely attributed to security forces, Amnesty said.
"We're looking at the situation with concern," President Barack Obama said Saturday in Johannesburg, South Africa. Those concerns extend to keeping U.S. embassies, consulates, diplomats and other personnel safe, he said.
"Everyone needs to denounce violence," Obama said, adding he would favor talks between Morsy and the opposition. "Nobody's benefiting now."
About 200 U.S. Marines in Sigonella, Italy, and Moron, Spain, have been put on alert as a precaution, according to two administration officials.
The U.S. State Department warned Americans to cancel all but essential travel to or within Egypt.
The United States has approved the departure of embassy and consulate employees and their dependents because of the unrest, a senior State Department official said. Nonessential employees and their relatives can get help leaving, if they choose to do so.
Pochter, the Kenyon College student from Maryland killed in Alexandria, wanted to improve his Arabic before returning to college in the United States. "He cared profoundly about the Middle East," his family said.
He was planning a career in the region in hopes of forwarding the peace process.