The Australian appeared on the embassy's balcony – the first time he has been seen for two months – and urged the US government to "reaffirm the revolutionary values it was founded on".
Mr Assange, who is wanted in Sweden for questioning on sexual assault allegations, thanked Ecuador for taking a "stand for justice" in giving him political asylum.
The decision has led to a diplomatic stand-off involving Ecuador, Sweden and the British Government, which insists it is legally obliged to hand him over.
Foreign Secretary William Hague has made it clear Mr Assange will not get safe passage out of the country.
Mr Assange, who enraged the US government in 2010 when WikiLeaks published tranches of secret US diplomatic cables, denies the allegations and fears being transferred to America if he travels to contest them.
Mr Assange thanked Ecuador and other helpful South American nations and his many supporters, plus his family, including his children "who have been denied their father". He said: "Forgive me, we will be reunited soon."
Earlier, his legal adviser Baltasar Garzon said Mr Assange had instructed his lawyers "to carry out a legal action" to protect his rights. He told media representatives: "Julian Assange has always fought for truth and justice and has defended human rights and continues to do so.
"He demands that WikiLeaks and his own rights be respected. Julian has instructed his lawyers to carry out a legal action in order to protect the rights of WikiLeaks, Julian himself and all those currently being investigated."
Mr Assange, who entered the building seeking asylum on June 19, thanked supporters who went to the embassy on Wednesday night, when it emerged that Britain had warned the Ecuadorian government that the diplomatic status of the embassy could legally be revoked.
He said: "Thank you for coming, thank you for your resolve, your generosity of spirit.
"On Wednesday night, after a threat was sent to this embassy, the police descended on this building. You came out in the middle of the night to watch over it, and you brought the world's eyes with you."