Thousands of Israelis have held protests against hate crimes, as the parents and brother of a Palestinian baby burned to death by suspected Jewish settlers continued to fight for their lives in hospital.

The firebombing of the family's home in the occupied West Bank that killed 18-month-old Ali Saad Dawabsheh on Friday, sparked an international outcry over Israel's failure to curb violence by hardline Jewish settlers.

The child's father, Saad, was being treated for third-degree burns at the Soroka hospital in southern Israel on Saturday, where a spokeswoman described his condition as "critical".

Mother Reham and four-year-old brother Ahmed were at Tel Hashomer hospital near Tel Aviv.

Family members told Al Jazeera that Reham's condition was life-threatening and had worsened since Friday.

The family's small brick and cement home in the village of Duma was gutted by fire early on Friday morning, and a Jewish Star of David spray-painted on a wall along with the words "revenge" and "long live the Messiah".

In Tel Aviv, around 2,000 people, including an uncle of the dead Palestinian child, participated in a rally on Saturday organised by the Peace Now settlement watchdog group.

"We call on the government to take strong action against the violence of the settlers and to restart immediately the peace process," Peace Now director Yariv Oppenheimer told the AFP news agency.

In Jerusalem, hundreds of people gathered to protest both the West Bank arson attack and an attack on Thursday, when an ultra-Orthodox Jew stabbed six participants of a Gay Pride march.

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"Flames of hatred have spread through our country, flames of violence, of hatred, of false and distorted beliefs," President Reuven Rivlin said in a speech to the crowd.

Smaller anti-intolerance rallies were held in other Israeli cities on Saturday night.

Meanwhile, in Duma, around 200 Palestinians marched on Saturday to protest the baby's death and were confronted by Israeli forces.

Medics told Al Jazeera that several protesters were treated for injuries from rubber bullets and tear gas.

On Saturday evening, hundreds of Gazans held a candlelight vigil and mock funeral for Dawabsheh.

A delegation of senior Palestinian officials, including Palestine Liberation Organization Secretary-General Saeb Erakat, Palestinian Authority security head Majed Faraj and Health Minister Jawad Awwad visited the Tel Hashomer hospital on Saturday.

Erakat, who is also chief negotiator for peace talks with Israel, reiterated that the Palestinians hold the Israeli government "fully responsible" for the attack on the family.