'Mariners who do business with Iran and Venezuela will face consequences from the United States of America,' Secretary of State Mike Pompeo said.
Yemen's government and the Southern Transitional Council agreed to a ceasefire proposed by the Arab coalition, calming recent escalation.
Hospitals in areas controlled by the militia are subject to restrictions and various 'taxes' as they struggle to fight COVID-19.
In recent years, the island has become a bone of contention between the Saudi-backed Yemeni government and UAE-trained separatists.
The resolution urges Iran to allow access to two sites in order to determine whether nuclear activity took place there in the early 2000s.
As the sponsor of the power-sharing agreement in Yemen, Saudi Arabia is responsible for ensuring its implementation, officials say.
The call comes in response to a UN report that says cruise missiles and drones used to attack Saudi oil facilities last year are Iranian in origin.
The International Atomic Energy Agency has expressed 'serious concern' that Iran has been blocking inspections at two sites.
The US has accused Moscow of deploying fighter jets in support of Khalifa Haftar, who has been fighting the UN-backed government.
The warplanes left Russia and first stopped in Syria, where they were repainted to camouflage their Russian origin before arriving in Libya, AFRICOM said.