Article 102 stipulates that the country's Constitutional Council shall meet and then propose to the parliament to declare the inability of the president to exercise his functions due to verified, serious and enduring illness.
The army chief called Tuesday for Bouteflika to be declared unfit to govern.
In response to Salah's address, the Algerian Constitutional Council said it would hold a special meeting, according to El Bilad TV.
"The solution that must be adopted to end the crisis and answer the legitimate demands of the Algerian people is the solution that guarantees the sovereignty of the constitution and the continuity of the state", Gaid Salah stressed.
Rarely does the military directly intervene in the country, but as thousands have been protesting against Mr Bouteflika's rule over the past month, the army has tried to restore order.
Protesters are pushing for an overhaul of the powerful establishment entrenched in power since independence from France in 1962, and the old guard hopes it can put forward a candidate approved by the army.
Gaid Salah, speaking on Tuesday, praised the maturity of Algerians and their peaceful protests.
Those same protests succeeded in forcing Bouteflika to publicly announce he would not be running for a fifth term; however, protests have not ceased and show no signs of letting up until the president steps down.
Critics said the postponement was unconstitutional and alleged it amounted to a desperate bid to cling to power until the president's entourage found a replacement who would continue corruption and other policies they are protesting.
Talks have been set up to oversee the country's political transition, draft a new constitution and set the date for elections.
The president also dismissed his prime minister and appointed a new one the same day.
However, Bedoui is struggling to form a government, with many potential ministers trying to keep their distance from the unpopular president.