In fresh clashes on Tuesday, Israeli forces killed two Palestinians, according to local reports.
An Israeli spokesman disputed claims that tear gas was the cause of death, saying "we have several accounts that question the validity of this statement".
The family prefers to focus on who fired the gas rather than the series of decisions that led to a baby being a few hundred metres from the Israeli border during the protests and clashes.
More violence was expected amid simmering tensions last night on the day that marked the 70th anniversary of the start of the first war between the newly created Jewish state and its Arab neighbours in 1948.
Israel has defended its response, blaming Hamas for fomenting violence and insisting it needed to protect nearby Jewish settlements from protesters getting through the border fence.
It was reported that when Leila began crying her uncle took her towards the protest area in order to locate his sister.
She was said to have then been victim of a teargas canister dropped by an Israeli drone.
Israeli troops used tear gas and live fire against protesters, as they approached the border fence, wounding nearly 3,000 and sparking condemnation from the United Nations (UN).
Close to the border he eventually found his mother Heyam and handed Leila over to her.
"I could barely breathe", she said.
The Palestinian Health Ministry said the soldiers killed Bilal Bodeir al-Ashram, 18, from the Nusseirat refugee camp, east of Deir al-Balah, in Central Gaza. 'We got away from the gas.
Leading the procession, her grief-stricken mother pressed the baby's body to her chest and cried: "Let her stay with me, it is too early for her to go". They often can not afford the medicines or medical bills for follow up care and extremely anxious about their children's futures - which they say were already being steadily eroded by more than 10 years of Israeli blockade, falling donor interest.
On Monday, tens of thousands of Palestinians rallied near Israel's border fence to protest a blockade of their territory and the move of the US Embassy to contested Jerusalem that day.
Mr Regev said: "If we allow the Hamas terrorists to enter Israel, we will have dead Israelis".
But during a UN Security Council meeting, United States ambassador Nikki Haley said: "No country in this chamber would act with more restraint than Israel has".