The spokesman told Ahram Online Gaber Salah died after suffering gunshot wounds to the head and neck.
"Revolutionaries went to the streets to commemorate their martyrs but they became martyrs themselves," the group said in a statement.
Salah's death reminded Egyptians of President Mohamed Morsi's promise to restructure the interior ministry and put former regime figures on trial for killing demonstrators.
Several parties and political groups have issued statements condemning the latest violence and expressing condolences for Salah's death.
"The use by police of live ammunition and rubber bullets targeting vulnerable body parts is a crime," the Constitution Party said.
The Adl Party and the Egyptian Current Party blamed the continuing violence on the lack of accountability for previous killings.
The clashes between police and protesters began Monday during a demonstration commemorating the 47 who died and thousands injured on Mohamed Mahmoud Street last year.
Putin thinks Obama would save him if he were drowning
Justin Bieber crashes Drake Bell's album release party