Hamas struck back on Monday by ordering the immediate shutdown of the clinics.
Doctors who do not comply will be fired, and clinics will also be scrutinised to ensure they are properly registered and licensed, Hamas officials said.
The face-off has largely paralysed Gaza's medical system, putting it at the mercy of the rivalry between Hamas and Fatah led by Mahmoud Abbas, the Palestinian president.
Hamas seized control of Gaza in June after routing Fatah forces, while Abbas formed a new government in the West Bank.
Despite the conflict, emergency services were still operating around the clock, officials said.
Meanwhile, the Palestinian government closed down 103 institutions in the West Bank and Gaza on grounds they violated laws governing civil groups.
Salam Fayyad, the prime minister, on Monday denied the move was a crackdown on Hamas alone, and said the closures were part of a wider reform.
"When there is a violation of the law, should I stop there and say there is a sensitivity because this institution might be a Hamas one? This is not acceptable," he said.
It was unclear how Fayyad's government would carry out the edict in Hamas-ruled Gaza, but the bank accounts of the targeted groups would be frozen, he said.
In other developments on Monday, Israeli troops shot dead a Palestinian man from across the border with the central Gaza Strip, officials said.
Palestinian medics and witnesses said that Farid Abu Dhaher, a farmer in his 40s, was working his land near the heavily fortified fence between Gaza and Israel when he was shot.
An Israeli army spokeswoman said troops opened fire at a man who looked as though he was trying to plant a bomb in the border area, although no explosive device had been found by late afternoon.
Meanwhile, in the central Gaza town of Khan Yunis, a 27-year-old Palestinian who was wounded two weeks ago during an Israeli incursion died of his injuries, a medic said.
He was named on Monday as Salim Asfur.
In the occupied West Bank on the other hand, an Israeli soldier who took a wrong turn into the town of Jenin was returned unharmed by Palestinian security forces.
The Israeli army said the soldier inadvertently drove into Jenin.
He was picked up on Monday by Palestinian security forces loyal to Abbas and escorted back to an Israeli position, the army said.
The Islamic Jihad group, however, said it had "abducted the soldier who had sneaked into the city", but its fighters were "surprised by members of the Preventive Security Service who surrounded them and took him".
A crowd overturned and torched the soldier's abandoned car, tearing pieces of metal off the vehicle after a fire truck doused the flames, TV footage showed.
Earlier in the day, Israeli fighter jets targeted a rocket launcher near the northern West Bank town of Beit Hanun but caused no casualties, the Israel army and witnesses said.
On the political front, Ehud Barak, Israel's foreign minister, said on Monday that Israel will not be able to carry out a major West Bank pullback for two and a half years because it first needs a missile defence system in place to protect it against Palestinian rocket fire.
"The things we see in Gaza do not allow us to change our actions in [the West Bank]," he was quoted as saying, referring to daily rocket fire at Israel from Gaza by Palestinian factions.
Barak's timetable - announced a day before a meeting between Israeli and Palestinian leaders and as preparations move ahead for a US-sponsored peace conference - cast doubt on chances of a quick breakthrough in stalled peacemaking.
Nevertheless, Ehud Olmert, the Israeli prime minister, and Abbas will meet in Jerusalem on Tuesday, for talks expected to focus on principles for Palestinian statehood, Olmert's office confirmed on Monday.