The meeting in Riyadh, where Esper arrived late on Monday after an unannounced visit to Afghanistan, also took in defence issues and the current situation in the region.
Saudi Health Minister Dr Tawfiq Bin Fawzan Al Rabiah thanked the Custodian of the Two Holy Mosques.
Saudi Arabia's King Salman has been admitted to a hospital in the capital, Riyadh, for medical tests due to inflammation of the gallbladder.
In a separate royal decree on Friday, Abdulaziz Al Duailej was appointed as head of the General Authority of Civil Aviation, as the kingdom prepares to end travel restrictions for its citizens and reopen its borders from May 17.
Sultan Haitham Bin Tariq Al Said of Oman visited Saudi Arabia on Sunday on his first official foreign trip since assuming power last year. Sultan Haitham arrived in the Red Sea city of Neom for a two-day visit.
The king "underwent a laparoscopic surgery to remove the gall bladder today... at King Faisal specialist hospital in Riyadh," the royal court said in a statement carried by SPA.
In a telephone call with US President Donald Trump, Saudi King Salman Bin Abdulaziz Al Saud affirmed the Kingdom’s keenness to reach a permanent and just solution to the Palestinian cause to bring peace, which he said is the main starting point for the Kingdom’s efforts and the Arab Peace Initiative, reported the Saudi Press Agency (SPA).
President His Highness Sheikh Khalifa Bin Zayed Al Nahyan has sent a cable of condolences to the Custodian of the Two Holy Mosques, King Salman Bin Abdulaziz of Saudi Arabia, on the death of Prince Bandar Bin Abdulaziz Al Saud.
In his message, Sheikh Khalifa also asked Allah Almighty to rest the deceased’s soul in Paradise and grant solace and patience to the Al Saud family.