The powerful Algerian army had called for the 82-year-old to be declared incapable of carrying out his duties.
He suffered a stroke six years ago and has rarely appeared in public since.
Leaders of the protests had rejected Mr Bouteflika's offer to go by the end of his current term - 28 April - as not enough.
The demonstrations have also led to calls for the whole political system, in which the military plays a significant role, to be overhauled.
The governing National Liberation Front (FLN), which postponed elections, had vowed to organise a national conference on reforms.
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