The council said in a statement on Sunday that it rejected "the candidates' applications" and subsequently declared "the impossibility of holding presidential elections on July 4".
It did not set a new date for the presidential election, asking interim President Abdelkader Bensalah to organise a vote at a later date, state television said.
OPINION What is next for Algeria after Bouteflika's resignation?
by Thomas Serres
The move will likely extend the rule of the interim president, who was meant to stay only until the vote to elect a new president after Bouteflika ended his 20-year rule in the wake of mass protests.
Bensalah, the former upper house speaker, had been appointed as interim leader until July 9.
Demonstrations have since continued, with protesters demanding Bensalah's resignation and an end to the dominance of the elite who have ruled Algeria since it won independence from France in 1962.
"Algerians have protested for 15 weeks, we are heading towards a sixteenth week of protests so the issue here is Bensalah has to listen to the protesters," said Mohamed Kirat, professor of media studies at Qatar University.
"Because with the current government and Bensalah and Prime Minister Noureddine Bedoui still in power, there is no way that Algerians will go to vote for a new president," added Kirat.
On Friday, hundreds of thousands again took to the streets of Algiers and other cities to call for Bensalah's removal and that of Prime Minister Bedoui, who was appointed by Bouteflika days before he stepped down.
The July vote had been backed by Algeria's army chief, General Ahmed Gaid Salah, who has been managing the transition. Salah, who earlier supported Bouteflika, has called for the impeachment of the long-time leader.
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