The police officer was shot in the chest from a rooftop while taking an arrested protestor into a police vehicle, Xinhua reported citing the statement.
About 50 members of the Muslim Brotherhood, from which Morsi hails, staged the protest at one of Maadi neighbourhood in southern Cairo.
Since Morsi's removal by the military last July, his supporters have been regularly staging such protests, especially on Fridays, Egypt's weekends, despite a massive security crackdown that left about 1,000 killed and thousands of others arrested over the past 10 months.
But recently, pro-Morsi protests were diminishing due to mounting security measures in the country.
Ex-military chief Abdel-Fattah al-Sisi, who led Morsi's removal, was elected president late last month. The new president won nearly 97 percent of votes in a landslide victory against his sole leftist rival Hamdeen Sabahy in a three-day election.