It comes after the nation issued a new set of rules for Assange, who has lived in its London embassy for years.
Wikileaks claim Mr Assange's access to the outside world has been "summarily cut off" and say Ecuador has threatened to remove the protection he has had since being given political asylum.
WikiLeaks said the government of Ecuador refused a visit by Human Rights Watch general counsel Dinah PoKempner, who has likened Ecuador's isolation to "solitary confinement", and had not allowed several meetings with his lawyers. The WikiLeaks founder sought the protection of Ecuador in 2012 to avoid extradition to Sweden, where he faced rape allegations, or the USA, where he could be punished for publishing secret government documents.
Ecuador feared Mr Assange could make more controversial statements that would have strained the country's relationship with the United Kingdom and the European Union.
Mr Garzon said Mr Valencia was named in the lawsuit because he served as the intermediary between Mr Assange and the Ecuadorean Government.
Ecuador this month created the new protocol governing his stay at the embassy.
At least 130 Gazans hit by Israeli gunfire, says ministry
Clashes, which have been repeatedly taking place since late March, intensify during every Friday prayers. Protesters set fire to tires to blur the vision of Israeli snipers deployed at the borders.
"It is not a comfortable situation, it is an inhuman situation in any case because the solution is being extended over time", Garzón had told reporters on Thursday in the Ecuadorian town of Ambato. His website, Wikileaks became famous after the publication of secret materials of the Pentagon, state Department and other government agencies of the United States. "The protocol makes Assange's political asylum contingent on censoring his freedom of opinion, speech and association", the organization continued.
Assange's attorneys filed papers Friday in a Quito court.
Ecuador granted him asylum in August 2012, saying it feared his human rights might be violated if he was extradited.
Mr Assange was granted refuge in the Ecuadorian embassy after he breached bail's terms received following his arrest two years earlier.
The WikiLeaks founder argued that the claims were "without basis" and said he feared it could lead to his extradition to the U.S. over WikiLeaks' publishing of classified information.
It was through Wikileaks in 2010, that U.S. Army veteran Chelsea Manning was able to leak classified data.