The two spoke in London at the end of talks with European and Middle Eastern allies seeking a multilateral approach in resolving the Syrian crisis, including a sustained cessation of violence and humanitarian aid deliveries to the beleaguered Syrian city.
Up to a quarter of a million people are trapped in Aleppo as the Syrian military, backed by Russian warplanes, pounds its streets into rubble.
Kerry called the bombing of Aleppo "a horrendous step back in time" and the greatest humanitarian disaster since World War II.
The secretary of state said Russia claims to be fighting terror in Syria, but he said that 80% to 85% of its bombing is directed at the moderate opponents of Syrian President Bashar al-Assad in the civil war.
"No one is fooled by this," Kerry said.
Admitting frustration at the slow pace of negotiations while so many people have died, Kerry said: "We are pursuing diplomacy because these are the tools we have."
Johnson said, "Our challenge to the Russians is to do the right thing for humanity," adding that sanctions and other diplomatic measures will "eventually come to bite the perpetrators."
Attendees at the London talks included the UK, the United States, France, Germany, Italy, Saudi Arabia, Qatar, Turkey, the United Arab Emirates, Jordan and the European Union, the British Foreign Office said.
Kerry flew to London after meeting Saturday in Lausanne, Switzerland
, with Russian Foreign Minister Sergey Lavrov; Staffan de Mistura, UN special envoy for Syria; and officials from regional powers with influence on battlefield outcomes in Syria.
No breakthrough occurred Saturday, but Kerry said the parties reached a "broad agreement" on some important points, specifically a "desired outcome on ending conflict," in his remarks to the press.
Tension between Russia and the United States is making peace in Syria harder to achieve.
Washington called off bilateral talks with Russia this month following the collapse of a short-lived ceasefire in Aleppo and Syria's renewed offensive against the city's rebel-held east.
The United States, France and Britain are among the Western powers to suggest in recent days that the Syrian regime and its Russian backers could be guilty of war crimes in Aleppo.
For his part, al-Assad told a Russian newspaper Thursday there was no other option but "to clean" Aleppo and use it as a "springboard" to push rebel forces out of Syria.
Meanwhile, a key development occurred Sunday in Syria, with ISIS losing control of a symbolic stronghold in the country's north near the Turkish border.
The Free Syrian Army, a Turkish-backed faction, took back the town of Dabiq from ISIS
, Turkish state media and a monitoring group said.
The Syrian Observatory for Human Rights, a UK-based organization, said it received reports that groups of ISIS fighters had withdrawn from Dabiq overnight.
In Iraq, a long-awaited operation to seize Mosul after two years of ISIS control
will begin in the coming days, militia groups warned Sunday.
The operation to liberate Mosul is expected to be a "messy and prolonged battle," CNN senior international correspondent Ben Wedeman said in the Iraqi city of Irbil, adding that the battle would likely come down to "street-to-street fighting."