Bilal Kayed has been refusing food for 56 days in protest at his detention without trial by Israel and was moved from prison to Barzelai hospital, in the southern city of Ashkelon, when his health deteriorated markedly.
The Physicians for Human Rights NGO says his vision is failing, he has difficulties standing and that doctors have warned he could be at risk of a stroke.
Palestinian officials say he has kidney problems.
On Tuesday Arab and some Jewish supporters of Kayed picketed the hospital, some carrying banners reading "Administrative detention is not legal," a reference to an Israeli practice allowing detainees to be held without trial for renewable six-month periods.
Arab Israeli MPs were among the protesters.
Right-wing Israelis, some shouting "death to terrorists", staged a counter-demonstration and the two sides exchanged punches, an AFP journalist at the scene reported.
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When police intervened, the right-wingers threw rocks at officers.
Police said they arrested 10 Jews and three Arabs suspected of public disorder, assaulting police or interfering with police in the performance of their duty.
They did not say if the Arabs arrested were Israeli citizens or Palestinians from annexed east Jerusalem, who have free access to Israel.
Kayed was to have been released in June after serving a 14-and-a-half-year sentence for activities in the Popular Front for the Liberation of Palestine, labelled a terrorist organisation by Israel, the European Union and the United States.
Instead, Israeli authorities ordered he remain in custody until further notice.
Israel says administrative detention allows authorities to hold suspects while continuing to gather evidence, while Palestinians, human rights groups and members of the international community have criticised the system.
Of more than 7,500 Palestinians currently in Israeli jails, about 700 are being held under administrative detention, Palestinian rights groups say.
Palestinians have regularly gone on hunger strike in protest at their detention.