Alix Didier Fils-Aimé said his priority was “restoring security”, according to AFP news agency.
Notwithstanding the country’s “difficult circumstances”, he promised to put all of his energy, skills and “patriotism at the service of the national cause”.
The businessman, who stood unsuccessfully for a seat in the Senate in 2015, studied at Boston University. He was installed by the country’s ruling council after the previous leader, Garry Conille, was ousted less than six months into the job.
The Spirit Airlines flight had been scheduled to land at Toussaint Louverture International Airport just before 12:00 (17:00 GMT) when it was hit.
Unverified video of the incident shared on social media appeared to show several bullet holes on the inside of the aircraft, where the crew sit during take-off and landing.
Spirit Airlines said that damage “consistent with gunfire” had been found when the plane was inspected at Santiago Airport. The aircraft was taken out of commission, Spirit added.
The airline said it had also suspended flights to Haiti “pending further evaluation”.
Two other US airlines, American Airlines and JetBlue, have also suspended flights to Haiti until at least Thursday.
The security situation further deteriorated in Haiti in recent months. In October, gang members opened fire at a UN helicopter, causing some airlines to temporarily cancel flights to the Caribbean nation.
A UN-backed policing mission, led by officers from Kenya, had begun in June in an attempt to wrest back control from gangs.
Spirit is a low-cost airline based in Florida which flies throughout the US, Caribbean and Latin America.