Mr Assange has been at Ecuador's London embassy since June but the UK has said it could lift its diplomatic status.
He faces extradition to Sweden over sexual assault claims, which he denies.
Ecuador's President Rafael Correa has suggested Mr Assange could co-operate with Sweden if assurances are given that there would be no extradition to a third country.
Supporters of Mr Assange - who on Sunday urged the US to end its "witch-hunt" against the Wikileaks site - claim he could face persecution and even the death penalty if sent there.
A document agreed at the Union of South American Nations meeting said it supported the country "in the face of the threat" to its London embassy.
After Ecuador's Foreign Minister Ricardo Patino finished reading the final declaration from the Union of South American Nations (Unasur) summit, he joined hands with his fellow foreign ministers from across South America and raised them aloft.
The BBC's Will Grant said it was a symbolic but important show of unity in a region which considers the UK government's approach over Mr Assange to have been colonialist and threatening.
Ecuador has described a letter from the British government drawing attention to the Diplomatic and Consular Premises Act 1987 as "intolerable" and an "explicit threat".
The act could allow the UK to lift the diplomatic status of Ecuador's embassy in London to allow police to enter the building to arrest Mr Assange for breaching his bail terms.