Before the Navy Seals there were the Underwater Demolition Teams (UDT). They assisted the recovery of the space capsules.
The Underwater Demolition Teams (UDT) were a special-purpose force
established by the United States Navy during World War II. They came
to be considered more elite and tactical during the Korean and Vietnam
Wars. Their primary WWII function began with the reconnaissance and
removal of natural or man-made obstacles on beaches prior to
amphibious landings. They later were assigned to assist in the
recovery of space capsules and astronauts after splash down in the
Mercury and Apollo space flight programs. The United States Navy's
Underwater Demolition Teams were pioneers in underwater demolition,
closed-circuit diving, combat swimming, and midget submarine (dry and
wet submersible) operations. Commando training was added making them
the forerunner to the United States Navy SEAL program that exists
The Navy moved forward to establish its new special operations force
and in January 1962 commissioned SEAL Team ONE in NAB Coronado and
SEAL Team TWO at NAB Little Creek. UDT-11 & 12 were still active on
the west coast and UDT-21 & 22 on the east coast. The SEALs quickly
earned a reputation for valor and stealth in Vietnam, where they
conducted clandestine raids in perilous territory. In May 1983, the
remaining UDT teams were reorganized as SEAL teams. UDT 11 became SEAL
Team Five and UDT 12 became Seal Delivery Vehicle Team One. UDT 21
became SEAL Team Four and UDT 22 became Seal Delivery Vehicle Team
Two. A new team, SEAL Team Three was established in October 1983.
So UDT and Navy Seals both existed between 1962 and 1983. I guess they did not want to replace a trained and experienced successful recovery team by another one without experience.
Note the letters UDT on the flotation collar around Mercury 7.
Image from a navy frogmen page.