Lebanon’s new Hezbollah-supported president is courting Saudi Arabia in an effort to restore billions in military aid, a delicate task as his country grows closer to Iran, the kingdom’s main rival for Middle East power.

Michel Aoun’s country has for years been a political battleground in the regional struggle between Iran, the backer of the Lebanese Shiite militant and political group Hezbollah, and Saudi Arabia, a supporter of the country’s main Sunni bloc.

The ascendancy of Mr. Aoun, Hezbollah’s main Christian ally, was a victory for the group—and a sign of the waning influence in Lebanon of Saudi Arabia, which had opposed his candidacy. […]