Al-Zubaidi, whose forces seized a key government building in Aden on Saturday, blamed guards at the presidential palace for flaring up a fight by opening fire on his supporters.
"It is heartbreaking that during Eid al-Adha families are mourning the death of their loved ones instead of celebrating together in peace", said Lise Grande, the top United Nations humanitarian official Yemen, in a statement.
Southern Yemen was an independent state until 1990 and the north is widely perceived to have imposed unification by force.
The Western-backed coalition said it attacked an area that poses a "direct threat" to the Saudi-backed government of Abd Rabbu Mansour Hadi, which is temporarily based in Aden.
The International Crisis Group think tank has warned that the Aden clashes risk creating a "civil war within a civil war".
He said the coalition called on the separatist Southern Transitional Council and its Security Belt paramilitary forces to "return immediately to its positions and withdraw from all positions they seized in the past days".
Fighting in the city flared on Wednesday after two members of the Security Belt were killed in clashes with other loyalist forces after the funeral of police personnel killed in the city last week, according to security officials.
Separatist commanders accused the Al-Islah Islamist party of killing one of its commanders and "infiltrating" the Hadi government, which is politically and financially supported by Saudi Arabia.
"What is happening in the temporary (government) capital of Aden by the Southern Transitional Council is a coup against institutions of the internationally recognised government", the foreign ministry said in a Twitter post.
The sheer number of Islamists, separatists, and former government forces on the side of the Saudi-backed invasion of Yemen has long made this a very complicated war.
The UAE has expressed "deep concern" over the continuation of armed confrontations in Yemen's Aden, calling for calm and de-escalation, official news agency WAM reported on Saturday.
Mr Hadi's office said the two men discussed the separatist "coup" against the government and "various other crimes against the sons of Aden". "We reject any use of arms in Aden and breach of security and stability", he said.
More than 70 people, including civilians, have been killed, AP news agency reported, citing Yemeni security officials.
Prince Khalid bin Salman, the vice minister of defense and a brother of Saudi Arabia's de factor ruler, Crown Prince Mohammed bin Salman, wrote on Twitter that "violence in Aden will create a situation that could be utilized by terrorist organizations like the Houthis, al-Qaeda/ISIS, which KSA will never condone". The UN says over 24 million Yemenis are in dire need of humanitarian aid, including 10 million suffering from extreme levels of hunger.