Hezbollah and Israel fought a two-month war last year that left the militant group badly weakened, though it retains part of its arsenal.
Israel has kept up its air strikes on Hezbollah targets despite a November ceasefire, and has threatened to continue them until the group has been disarmed.
In a speech on Thursday, Aoun said Beirut was demanding "the extension of the Lebanese state's authority over all its territory, the removal of weapons from all armed groups including Hezbollah and their handover to the Lebanese army".
He added it was every politician's duty "to seize this historic opportunity and push without hesitation towards affirming the army and security forces' monopoly on weapons over all Lebanese territory... in order to regain the world's confidence".
Under the November ceasefire, Hezbollah was to withdraw its fighters north of the Litani river, about 30 kilometres (20 miles) from the Israeli border.
Israel was meant to withdraw all its troops from Lebanon, but has kept them in five areas it deems strategic.
The truce was based on a two-decade-old UN Security Council resolution that said only the Lebanese military and UN peacekeepers should possess weapons in the country's south, and that all non-state groups should be disarmed.
However, that resolution went unfulfilled for years, with Hezbollah's arsenal before the latest war seen as far superior to the army's, and the group wielding extensive political influence.
Aoun took over the presidency in January ending a two-year vacancy -- his election by lawmakers made possible in part by the shifting balance of power in the wake of the conflict.
On Wednesday, Hezbollah chief Naim Qassem said that "anyone calling today for the surrender of weapons, whether internally or externally, on the Arab or the international stage, is serving the Israeli project".
He accused US envoy Tom Barrack, who has visited Lebanon several times in recent months, of using "intimidation and threats" in his talks with senior officials with the aim of "aiding Israel".
- Collapse or stability -
Israel has carried out near daily strikes in Lebanon in recent months, targeting what it says are Hezbollah militants and infrastructure, but the group has refrained from striking back.
Israel launched several strikes on Hezbollah strongholds in the south and east on Thursday, targeting what it said were sites used by Hezbollah to manufacture and store missiles.
Defence Minister Israel Katz said the targets included "Hezbollah's biggest precision missile manufacturing site", and the military said it had hit "infrastructure that was used for producing and storing strategic weapons" in south Lebanon and the Bekaa Valley.
In his speech, Aoun said Lebanon was at "a crucial stage that does not tolerate any sort of provocation from any side".
"For the thousandth time, I assure you that my concern in having a (state) weapons monopoly comes from my concern to defend Lebanon's sovereignty and borders, to liberate the occupied Lebanese territories and build a state that welcomes all its citizens," he said, addressing Hezbollah's supporters as an "essential pillar" of society.
Lebanon has proposed modifications to "ideas" submitted by the United States on Hezbollah's disarmament, Aoun added, and a plan would be discussed at a cabinet meeting next week to "establish a timetable for implementation".
Aoun also demanded the withdrawal of Israeli troops, the release of Lebanese prisoners and "an immediate cessation of Israeli hostilities".
"Today, we must choose between collapse and stability," he said.
Hezbollah is the only group that held on to its weapons after Lebanon's 1975-1990 civil war, doing so in the name of "resistance" against Israel, which occupied south Lebanon until 2000.
Lebanon has also committed to disarming Palestinian militant groups that control the country's refugee camps.
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