Fighting has claimed at least 60 lives in recent weeks
In a statement Mr Abbas - the Fatah leader - said some Hamas figures were "planning a coup against the legitimate [Palestinian] institutions".
Since Monday at least 16 people have died in fighting between rival groups.
Earlier, Hamas officials accused Fatah of trying to assassinate PM Ismail Haniya, by firing a rocket-propelled grenade at his home.
The building was damaged but no-one was injured.
This was the third time Mr Haniya had come under fire since Monday.
The political and military leaders of Hamas are planning a coup against the legitimate institutions, thinking they will be able to control the Gaza Strip by force
Mahmoud Abbas statement
Mortars were also fired at the Gaza compound of President Mahmoud Abbas. He was not there at the time.
In another development, Hamas has declared parts of northern Gaza a military zone and barred Fatah members from the streets.
Hamas and Fatah on Monday agreed their seventh truce in just over a month to try to end the fighting that has claimed at least 70 lives in recent weeks.
However, the BBC's Aleem Maqbool says the ceasefire has been followed only by an upsurge in the killings, and in their brutality.
In one incident, the home a Fatah activist was attacked and three members of his family killed, including a 75-year-old woman and a 13-year-old boy.
A senior militant from the al-Aqsa Martyrs' Brigades, linked to Fatah, was also killed.
Jamal Abu al-Jedian was shot 41 times in his hospital bed in Beit Hanoun. His brother was also later found dead, Fatah said.
"Everybody is shooting at everybody," said one doctor quoted by Reuters.
A Hamas mosque preacher, Mohammed al-Rifati, was shot dead by Fatah gunmen.
Leaders attacked
On Tuesday morning, Palestinian political leaders became a target.
Mortars were fired at the compound of Mr Abbas, who leads Fatah, in Gaza City. He was not there at the time.
They crossed all the red lines
Hamas spokesman Fawzi Barhoum
Earlier, gunmen fired a rocket-propelled grenade at the home of his Mr Haniya, a leader of the rival Hamas movement, in the Shati refugee camp on the outskirts of the city.
A spokesman for Hamas said the house had been damaged, but that Mr Haniya and his family had not been harmed.
Fawzi Barhoum accused Fatah supporters of attempting to assassinate the prime minister.
"Hamas has decided to punish the attackers and the killers and it will not be reluctant to punish them without any mercy," Mr Barhoum said.
"They crossed all the red lines."
In two separate incidents on Monday, gunmen fired upon Mr Haniya's house and on a government building where he was holding a cabinet meeting.
No-one was hurt in the shootings but Mr Haniya was forced to suspend the cabinet session and ministers had to move elsewhere, an official said.
Our correspondent says the attacks are an indication of how little respect is being shown to the officials, who have been promising an end to the violence, from the moment it began over a month ago.
Meanwhile, the Israeli military carried out an air strike in northern Gaza, attacking what it said was a site near Beit Hanoun used by Palestinian militants to launch rockets into Israel.
The strike came shortly after militants fired rockets into the southern Israeli town of Sderot, wounding three people.
Islamic Jihad claimed responsibility for launching the rockets.
On Sunday, Israeli Prime Minister Ehud Olmert said his country's military action in Gaza would continue until militants stopped their attacks.
Mapa de la Franja de Gaza.
Los reportes informan que unos 200 miembros de la fuerza de seguridad de Hamás tomaron control del cuartel general de la organización Al Fatah en Gaza. También indican que en el ataque murieron 10 personas y por lo menos 30 más resultaron heridos.

"Hamás ha decidido castigar a los atacantes y asesinos, y estarán dispuestos a castigarlos sin misericordia. Cruzaron todas las líneas rojas" dijo Fawzi Barhoum, un alto vocero de Hamás durante las declaraciones realizadas a la prensa.

Proyectiles, misiles y cohetes fueron arrojados al cuartel antes de que las fuerzas insurgentes ingresaran a tomar el control.

El presidente palestino Mahmoud Abbas ha condenado el ataque y ha acusado a Hamás de intentar derrocar al gobierno palestino.

"Los líderes políticos y militares de Hamás están planeando un golpe contra las instituciones legítimas, pensando que podrán controlar la Franja de Gaza por la fuerza", dijo Abbas en una rueda de prensa.