A letter signed by 15 physicians warns that if attacks on medical facilities continue at their present rate, there could be none left within a month.
It calls on Mr Obama to impose a no-fly zone over Aleppo to stop air strikes.
Russia has meanwhile said its forces will hold fire for three hours each day from Thursday to let aid into Aleppo.
All military action, air and artillery strikes would be halted between 10:00 (07:00 GMT) and 13:00 (10:00 GMT), a defence ministry official told a briefing in Moscow.
However, the United Nations immediately said three hours would be insufficient to help the millions in need and appealed for 48-hour pauses.
Fighting has escalated in recent days, with rebels severing the government’s main route to the west of the city.
The offensive sought to break a siege by pro-government forces, who encircled the east in July with the support of Russian aircraft.
The letter was signed by 15 of the doctors remaining in eastern Aleppo – an earlier statement sent to the media had said there were 29 signatories.
The letter says that five years into the uprising against President Bashar al-Assad, doctors have “borne witness as countless patients, friends and colleagues suffered violent, tormented deaths”.
“The world has stood by and remarked how ‘complicated’ Syria is, while doing little to protect us. Recent offers of evacuation from the regime and Russia have sounded like thinly veiled threats to residents – flee now or face what fate?”
They say that in the past month there have been 42 attacks on medical facilities in Syria, 15 of them on hospitals where they work.
“Two weeks ago, four newborn babies gasping for air suffocated to death after a blast cut the oxygen supply to their incubators. Gasping for air, their lives ended before they had really begun,” they recall.
The doctors note that they have taken a pledge to help those in need and ask President Obama to “do your duty as well”.
“We do not need tears or sympathy or even prayers: we desperately need a zone free from bombing over eastern Aleppo to stop the attacks, and international action to ensure Aleppo is never besieged again.”
On Monday, the UN said countless civilians had been killed or injured over the past few weeks in the city, and that the targeting of hospitals and clinics had continued unabated.
Moreover, attacks on civilian infrastructure had left more than two million people without electricity or access to the public water network for several days, it added.
The UN said it was ready to assist civilians in Aleppo, but that it required a fully-fledged ceasefire or weekly 48-hour humanitarian pauses to reach them and replenish their food and medicine stocks, which were “running dangerously low”.