"Whoever gives this criminal regime an opportunity to kill tens of thousands more Syrians and destroy what is left of Syria does not want to recognise the extent of the tragedy," an SNC statement said.
Lakhdar Brahimi told the BBC that he was "not in a position to say yet" but was "committed to finding a solution".
Mr Brahimi, a former Algerian foreign minister, last week succeeded Kofi Annan, who resigned after both sides largely ignored his peace plan.
On Sunday, UN observers ended their mission to verify its implementation.
Their departure came after the UN Security Council agreed to allow their mandate to expire at midnight, and instead set up a new civilian office in Damascus to pursue political contacts which might lead to peace.
Meanwhile, Turkish Foreign Minister Ahmet Davutoglu has said his country is struggling to cope with the influx of refugees from Syria.
Mr Davutoglu again proposed that the UN establish a safe zone inside Syria to house those fleeing the conflict because Turkey would be unable to accommodate more than 100,000 people. To date, 70,000 have fled there, most within the past month.
Syrian opposition groups have also called for safe havens, but the idea has gain little traction internationally, with Russia and China opposed.
Since being confirmed as the new UN and Arab League envoy to Syria, Mr Brahimi has acknowledged that he has no concrete ideas of how to end the conflict in Syria, which he believes has been a civil war for some time.
On Monday, he told the BBC that he was not ready to say whether President Assad should step down despite widespread international condemnation of his government's crackdown on dissent since protests erupted in March 2011.
"I am not in a position to say yet, because I was appointed a couple of days ago. I am going to New York for the first time to see the people who I am going to work for, and I am going to Cairo see the Arab League," he explained.