Mr Aoun said “nothing justified” Saad Hariri’s continued absence, and that it was a breach of his human rights.
He added that he considered it an “act of aggression” against Lebanon.
For his part, Mr Hariri once again insisted on Twitter that he would soon return to Lebanon and that he was fine.
On Sunday night, he said in a television interview that he was free to leave Saudi Arabia and that he had left Lebanon in order to protect himself.
The Saudi government has denied holding him against his will or putting pressure on him to resign in an attempt to curb the influence of its regional rival, Iran.
The Shia Islamist Hezbollah movement, an Iranian proxy that Riyadh considers a terrorist group, is part of the unity government that Mr Hariri formed last year.
“Nothing justifies the failure of Prime Minister Hariri to return for 12 days. Therefore, we consider him to be held and detained, in violation of the Vienna Convention and human rights law,” President Aoun’s official Twitter account quoted him as saying.
“No decision can be made on a resignation from abroad,” he added. “He should return to Lebanon to present his resignation or withdraw it, or to discuss the reasons for it and how to address them.”
“We cannot wait longer and lose time. Affairs of state cannot be stopped.”
Mr Aoun is a Maronite Christian former army commander who is an ally of the Islamist militia and political party Hezbollah.
“This affects the dignity of all Lebanese,” he said, according to several Lebanese media outlets. “Hariri and his family are restricted in freedom of movement and are under surveillance.”
Will Saudi Arabia go to war with Iran?
Mr Hariri responded swiftly to the president’s comments, writing on Twitter: “I want to repeat and affirm that I am perfectly fine and I will return, God willing, to dear Lebanon as I promised you, you’ll see.”
A member of Mr Hariri’s political party, the Future Movement, also told Reuters news agency that the prime minister and his family were not being detained.
The BBC’s Martin Patience in Beirut says Mr Aoun was merely saying what many senior Lebanese officials have expressed privately. But, our correspondent adds, the fact the president has gone public shows that he feels he has got little to lose.
Mr Aoun had previously only questioned the “mysterious” circumstances of Mr Hariri’s resignation, which was made in a televised address from the Saudi capital.
Mr Hariri accused Iran of sowing “discord, devastation and destruction” in the region and said he sensed there was an assassination plot against him.
His father Rafik – himself a former Lebanese prime minister – was killed in a suicide bombing in Beirut in 2005. Several members of Hezbollah are being tried in absentia at a UN-backed tribunal at The Hague in connection with the attack, though the group has denied any involvement.
Mr Hariri, a Sunni Muslim who became prime minister for the second time in late 2016 in a political compromise deal that also saw Mr Aoun elected president, has close ties to Saudi Arabia.
