Jund Ansar Allah is one of five groups in the Gaza Strip that subscribe to the Salafist version of Sunni Islam, an ultra-conservative strain of the religion.
A five-minute video posted online that could not be verified showed images of attacks on the Islamic State group by what it called Crusaders, or Westerners, and showed images of the bomb it said was used in Friday's attack.
A commentary said "may God help... whoever manages to kill an American or a European from states participating in the coalition against the Islamic State, particularly the French who are full of hatred."
A source connected with the center said Friday that one or two blasts had damaged the south wall of the compound as well as some of the facade of the building, which also houses a French consular office.
On October 8, the center's offices were set ablaze after the apparent explosion of two faulty fuel tanks.
Police at the time suggested the explosion may have been a criminal act, but the results of their investigation were never released.
French diplomatic missions and official buildings have been on high alert since the start of a spate of kidnappings and beheadings of foreigners by Islamic State jihadists and their allies, particularly that of Frenchman Herve Gourdel in Algeria.
France has angered the radical Islamist group by taking part in air strikes on the group's forces in Iraq.