They accuse opponents of the coup of trying to destabilize the country, and have waged a security crackdown against members of the Muslim Brotherhood, from which Morsi hails, and other allies, arresting hundreds.
Pro-military media and state TV have frequently described the crackdown on Morsi supporters as a "fight against terrorism."
Morsi supporters have held near-daily protests since the coup but deny they are violent. After a bloody dispersal of their major sit-ins earlier this month, however, some retaliated by attacking police stations, government buildings and churches.
In Sinai, violence has only worsened since Morsi's overthrow.
The militant arrested there Saturday is suspected of leading an al-Qaida-linked group in an ambush where 25 off-duty policemen were lined up and shot last week, security officials said. The attack was one of the area's worst militant strikes on security forces.
The man, Adel Mohammed, also known as Adel Habara according to one official, has already been sentenced to death in absentia for killing soldiers in the Nile Delta last year.
Habara's arrest could potentially undermine militant activities in the area, where over two dozen security men have been killed alone since July. The security official said two other suspects were arrested along with Habara.
Authorities have been engaged in a long-running battle against militants in the northern half of the strategic region, which borders the Gaza Strip and Israel.
Earlier Saturday, security officials said 31 suspected militants have been arrested since Thursday, including two caught seeking treatment for wounds sustained in clashes with police.
In other violence nearby, a riot police officer was shot in the chest Friday while on patrol in the city of El-Arish, the capital of North Sinai governorate, the Interior Ministry said Saturday.
According to one security official, four militants have also been killed since Thursday. All officials spoke on condition of anonymity because they were not authorized to speak to the media.
Since the ouster of Morsi, his supporters have been organizing regular protests calling for his return. Several thousand took to the streets on Friday in scattered protests in different parts of Egypt, though demonstrations have largely waned in recent days.
In a sign authorities are responding to the easing of the pro-Morsi rallies, the government once again shortened a military-imposed curfew by two hours, making the evening lock-up in nearly a dozen Egyptian provinces only seven hours.
However, the full 7 p.m. to 6 a.m. curfew will remain in place for Fridays, when supporters of Morsi usually organize large rallies. Hundreds of his supporters, including leading Brotherhood members, have been arrested in a crackdown on the group.
On Saturday, authorities arrested another top figure from the Brotherhood in the Mediterranean city of Alexandria. Former lawmaker Sobhi Saleh was taken into custody on charges of inciting violence and disrupting public order, according to security officials.
The son of another Brotherhood figure, Saad Emara, was also detained and ordered held for 15 days, pending investigations for charges of inciting violence.
Also on Saturday, Health Ministry spokesman Khaled el-Khateeb raised the death toll in violence during Friday's pro-Morsi protests in several cities to eight. El-Khateeb said 221 were also injured most of them in clashes between Morsi supporters and local residents.
Those killed on Friday included one police officer and one civilian shot dead in a drive-by shooting targeting a police station in an upscale neighborhood of Cairo.
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Associated Press Writer Ashraf Sweilam in el-Arish, Egypt contributed to this report
