The Foreign Secretary wants demonstrations to take place outside Russia's embassy over Moscow's involvement in the Syrian conflict.
Boris Johnson warned that the "wells of outrage are growing exhausted" as the city of Aleppo faces destruction.
MPs have claimed Russia is responsible for committing war crimes in Syria, with an emergency Commons debate hearing calls for Britain to review its relationship with President Vladimir Putin's regime and seek tougher trade sanctions against it.
Mr Johnson told the Commons: "It is the UK week after week that is taking the lead together with our allies in America and in France, all the like-minded nations, in highlighting what is happening in Syria to a world where, I'm afraid, the wells of outrage are growing exhausted."
He added: "There is no commensurate horror, it seems to me, amongst some of those anti-war protest groups.
"I'd certainly like to see demonstrations outside the Russian embassy.
"Where is the Stop The War Coalition at the moment? Where are they?"
Mr Johnson added: "I believe it's up to us in the Government to show a lead and week after week in the UN we are indeed doing what we can to point out what the Russians are up to and to build up an international understanding of what is going on in Syria and I believe we're having some effect."
His call came after Labour's Ann Clwyd urged those who care about the plight of Syrian civilians to gather outside Russian embassies across the globe until the country stops its bombing campaign.
The Cynon Valley MP who served as special envoy on human rights in Iraq from 2003 until 2010, said worldwide protests would make it "crystal clear" to Russia and the regime of president Bashar Assad that "we think their actions are deplorable".
She said: "We need to speak up for and on behalf of our common humanity.
"So I would therefore call once again on everyone who cares about the plight of Syrian civilians to picket the Russian embassy in London and in capitals around the world from today.
"Two million, three million, four million people. It can be done. It has been done in the past.
"That should carry on until the bombing campaign stops and all the relevant players are forced to get around the table to end this horrible war."