Morsi, the candidate from the Muslim Brotherhood's Freedom and Justice Party, was declared the winner of a runoff election for president last weekend, besting former Prime Minister Ahmed Shafiq.
Morsi would be the first democratically elected president in Egyptian history.
Abdel Moneim Aboul Fotouh, who ran as an independent candidate in the first round of elections early this year, said the role of the future president is unclear.
"The president has now entered his presidential palace without knowing what his mandate is," he was quoted by Egyptian news service al-Ahram as saying.
The Supreme Council of Armed Forces took a series of steps before the weekend's developments that curtail the role of the president. The country doesn't have a post-revolution constitution in place that spells out presidential duties.
The Muslim Brotherhood last year stripped membership from Fotouh, saying his decision to run for president violated the group's decision to not field a candidate. He later decided to run as an independent candidate. The Islamist movement later reversed course by nominating Morsi as its candidate.