French President Emmanuel Macron, who had pressed Lebanon's fractious politicians to reach a consensus over naming Adib on Aug. 31, will speak about the crisis later on Sunday.
Patriarch Bechara Boutros Al-Rai, leader of the Maronite church, Lebanon's biggest Christian community, said Adib's resignation had "disappointed citizens, especially the youth, who were betting on the start of change in the political class".
Many top politicians, both Christian and Muslim, have held sway for years or even decades. Some are former warlords.
Rai said Lebanon now had to navigate "multiple dangers" without a government at the helm.
Rai's comments were echoed on the streets of Beirut, where mass protests erupted in 2019 as years of mismanagement, corruption and mounting debts finally led to economic collapse.
"We need new people. We need new blood," said Hassan Amer, 24, serving coffee at a roadside cafe in the capital, which was hammered by a huge port blast on Aug. 4 that killed almost 200 people.
Frustration at Adib's failure to form a government was voiced by many across religious communities.
A senior Shi'ite Muslim cleric, Sheikh Ahmed Qabalan, said it was a "disaster" that Adib had resigned and called for national unity, state news agency reported.
"We don't want sectarian or confessional talk," Lebanon's top Sunni Muslim religious leader, Grand Mufti Sheikh Abdul Latif Derian, was quoted by broadcasters as saying.
He said Lebanon's communities needed to show "understanding and balance" to face the major challenges ahead.
The cabinet formation effort stumbled after Lebanon's two main Shi'ite groups, Amal and the heavily armed Iran-backed Hezbollah, demanded they name several ministers, including finance, a key role as the nation draws up a rescue plan.
Saad al-Hariri, a former prime minister and leading Sunni politician, said he would not be involved in naming any new premier and that the French plan was "the last and only opportunity to halt Lebanon's collapse".
A French roadmap lays out a reform programme for a new government to help trigger billions of dollars of international aid. (Writing by Raya Jalabi; Editing by Edmund Blair and Nick Macfie) ((Raya.Jalabi@thomsonreuters.com;))