US Secretary of State John Kerry insisted Washington's decision to halt talks did not mean it was "abandoning the pursuit of peace".
And the UN rights chief called for action to halt the "ghastly avalanche of violence" unfolding in Syria's second city, which is reeling from some of the most brutal fighting in the five-year conflict.
The Syrian army announced a major Russian-backed military push nearly two weeks ago to capture the rebel-held eastern half of Aleppo, once the country's commercial hub.
On Tuesday, loyalist fighters seized several high-rise buildings from rebel groups in the city centre, pushing north towards other opposition districts.
The Syrian Observatory for Human Rights, a British-based monitor, said regime forces were "gradually advancing" after street battles on the front line dividing the rebel-held east from the government-controlled west.
"They are focusing on the tall buildings, which were once government administration buildings, because they can monitor entire streets and neighbourhoods from there," Observatory head Rami Abdel Rahman told AFP.
AFP's correspondent in Aleppo said fierce bombardment on the southern edges of the city could be heard throughout the night, although it was quieter by dawn.
Syrian state news agency SANA reported that rebel shelling on the government-held west, including on the Aleppo University campus, left six people dead on Tuesday.
Russia's foreign ministry said its embassy compound in Damascus had been hit in mortar shell fire on Monday, causing damage but no casualties.
'Patience runs out'
More than 300,000 people have been killed since the conflict erupted in March 2011, and the latest attempt at securing a diplomatic solution to the war has fallen flat.
Washington announced late Monday that it would suspend joint efforts to reinstate a nationwide truce, accusing Moscow of abetting strongman Bashar al-Assad's assault on Aleppo.
"Everybody's patience with Russia has run out," White House spokesman Josh Earnest told reporters.
A US official said US Secretary of State John Kerry is "laser-focused" on finding a diplomatic solution, but his near-daily communication with Russian counterpart Sergei Lavrov on the crisis is over.
Kerry said Tuesday the decision was one "we did not come to lightly."
"We are not abandoning the pursuit of peace, we are not going to leave the multilateral field, we are going to continue to try to find a way forward in order to end this war," he added, offering fierce criticism of Moscow.
"People who are serious about making peace behave differently from the way Russia has chosen to behave," he said.
Lavrov however said Tuesday: "We are not shirking our responsibility but consider that the crisis can only be resolved collectively."
The US-Russia truce plan for Syria had envisioned an end to hostilities, increased aid deliveries to besieged populations, and eventual coordination between rivals Moscow and Washington against jihadists.
But the truce collapsed after a week, with Russia blaming Washington for failing to convince rebels to distance themselves from jihadist fighters.
Russia and the US will keep a communications channel open solely to ensure their separate anti-jihadist bombing campaigns do not get in each other's way.
UN call for action
UN rights chief Zeid Ra'ad Al Hussein decried the "ghastly avalanche of violence and destruction" in east Aleppo, saying 100 children had been killed there in the past 10 days.
He urged the Security Council to introduce a limit on its members' veto power, to prevent countries like Russia blocking the referral of Syria's conflict to the International Criminal Court in The Hague.
On Monday, the largest hospital in the rebel-held side of Aleppo was completely destroyed in an aerial attack, according to the Syrian American Medical Society, which supports the facility.
Only five hospitals remain operational for the estimated 250,000 people living under a crippling government siege in east Aleppo.
In northeast Syria, a suicide attack hit a wedding late Monday as the bride and groom were exchanging their vows, killing 34 people, the local Kurdish government said.
The Islamic State (IS) group claimed the attack, saying that one of its members had fired on a gathering near Hasakeh city before blowing himself up, though it did not mention a wedding.
The bombing left rows of seats covered in blood at the wedding hall in Hasakeh province.
"I was taking pictures of the party, and all of a sudden I felt a huge explosion," said wedding photographer Walid Mohammad.
"I saw so many people die -- small kids, old people."