At least 38 people died on Wednesday.
The recent escalation in violence has killed at least 426 people and injured hundreds more, the Syrian Observatory for Human Rights monitoring group says. The city is home to about 400 thousand people.
An air strike warning system run by the Syrian Civil Defence, a rescue service in opposition areas, was by Wednesday afternoon sending alerts every few minutes, triggered when warplanes are spotted taking off from air bases.
Others held placards saying "Russians, you are killing Ghouta's children, but history records" and "Children are dying of hunger in under-siege Eastern Ghouta". They fear the terror of the bombardment will bring her into labour early, he said.
Over three million Syrian refugees live in Turkey including almost 540,000 in Istanbul after fleeing the conflict which began with anti-government protests in 2011.
"How much cruelty will it take before the worldwide community can speak with one voice to say enough dead children, enough wrecked families, enough violence, and take resolute, concerted action to bring this monstrous campaign of annihilation to an end?" he added. "There is no food, no markets", he said.
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And Panos Moumtzis, the United Nations humanitarian coordinator for Syria, said most of eastern Ghouta is without food, water or electricity.
Last week, Russian Federation negotiated an amendment that specified the ceasefire would not apply to the Islamic State and Al-Qaeda, which would allow the Syrian government to continue to fight against the groups.
Mrs May said she was "appalled by the escalation of air strikes in Eastern Ghouta and deeply concerned by reports of the ongoing deliberate targeting of civilians and civilian infrastructure, in blatant violation of worldwide humanitarian and human rights law".
"The Security Council has not been effective in addressing the Syrian situation", Ban said. Russian Federation has cast 11 vetoes on possible Security Council action on Syria since its civil war began in 2011.
The UN Security Council is set to vote later today on a draft motion for a ceasefire for Syria.
Medical charities say jets have hit more than a dozen hospitals, making it near impossible to treat the wounded.