Tillerson is also due to meet Parliament Speaker Nabih Berri and Prime Minister Saad al-Hariri.
Aoun urged the United States to "work on preventing Israel from continuing its assaults on Lebanese sovereignty" by land and sea, a statement from the presidency said.
Israel has recently escalated its threats over Lebanon's invitation for offshore gas exploration bids along the disputed maritime border.
"We support a free, democratic Lebanon free of influence of others, and we know that Lebanese Hezbollah is influenced by Iran".
The heavily armed Shi'ite movement is part of the Lebanese coalition government led by Sunni Muslim politician Hariri, under a complex political system that promotes sectarian power-sharing.
From 2006 until assuming his role in U.S. President Donald Trump's administration in 2016, Tillerson was ExxonMobil's chairman and CEO.
"We have never given heavy arms to the YPG so there is none to take back", Tillerson said.
Tillerson's talks in Beirut focused on developments in the region in addition to several issues related to Lebanon and its latest tension with Israel.
Yet Lebanese officials have continuously rejected this proposal over the years, arguing that the totality of the 860-sq km falls under Lebanon's jurisdiction.
U.S. Secretary of State Rex Tillerson is urging Hezbollah to cease its activities overseas to help reduce tensions in the region.
Tillerson arrived in the Lebanese capital earlier in the day as part of a regional tour.
The policy the three leaders wish to put forth is one hinging on the notion that Israel's actions along the southern and maritime border infringe upon Lebanon's rights.
The long-standing dispute resurfaced recently as Lebanon invited companies to sign exploration deals.
Israel claims that Lebanon will be drilling in areas owned by Israel. USA officials have previously tried to mediate the Israeli-Lebanese dispute.