The U.S., Australia, India and Japan grouping, often referred to as the quad, continue to have regular diplomatic meetings.
The Joint Chiefs of Staff approved the medal for Operation Wild Boar, a three-day mission to rescue a soccer team.
Seven nations in addition to Thailand and the United States are active participants.
South Korea and the United States signed a new cost-sharing plan, which requires South Korea to pay about $924 million.
The Navy's 7th Fleet said the MH-60 Seahawk crashed shortly after takeoff while the carrier was off the Philippine coast.
Derek Anderson, a rescue specialist with the Air Force based in Okinawa, said the dozen boys were "incredibly resilient."
A team of 42 U.S. military personnel assisted in the Thai cave rescue.
All 12 boys and their soccer coach have been rescued from a flooded cave in northern Thailand, the Thai navy SEALs said.
The U.S. military is assisting Thai-led efforts to rescue members of a soccer team and their coach who are trapped in a cave.
The operation around Tham Luang Nang Non cave suffered its first death when a former Thai navy SEAL passed out underwater.