"This is an illegal and unconstitutional step because Abu Mazen's [Abbas's] tenure is over and he has no right to issue any decree concerning this [election]," Fawzi Barhoum, a Hamas spokesman, was quoted by the AFP news agency as saying.
He said the move by Abbas, whose presidential term expired in early 2009, was a "deliberate attempt to make [Palestinian] divisions permanent".
"Abbas has succumbed to American pressures not to reconcile with Hamas unless we recognised the Quartet conditions"
Sami Abu Zuhri,
But an official close to Abbas said there was still time to avoid an open clash.
"If we reach to an agreement, the election date can be rescheduled," he said.
Nour Odeh, Al Jazeera's correspondent in Ramallah, said: "The January 24 date was agreed on by all factions in Cairo. This is not a date that someone has pulled out of their pocket.
"However, Egypt [which is mediating the unity talks] had asked that these elections be postponed untill June, and that was agreed to by all parties provided that all factions sign the unity deal that was presented."
Fatah has approved the Egyptian-mediated deal, but Hamas is yet to sign it.
But Sami Abu Zuhri, another Hamas spokesman, said that Friday's decree was "a fatal blow to reconciliation efforts".
"Abbas has succumbed to American pressures not to reconcile with Hamas unless we recognised the Quartet conditions," he said.
Marwan Bishara, Al Jazeera's senior political analyst, said that if elections were to go ahead in January he expects to turnout to be limited.
"My fear is that they will only be held on the West Bank and less half of the people in Gaza and the West Bank will end up participating," he said.
"This will put quite a dampener on their legitimacy as democratic elections among the Palestinians."
An opinion poll earlier this month showed Abbas losing popular support, mainly over his response to a UN report criticising Israel for alleged war crimes in Gaza last January.
The poll indicated that Abbas and Ismail Haniyeh, Gaza's Hamas leader, were about even in popular support, although Fatah was still ahead.
Hamas won Palestinian elections in January 2006, but many world powers refused to recognise their government.