Navy and Marine Corps divers found the remains in the sealed compartments aboard the McCain, said Swift, commander of the Pacific Fleet.
The Royal Malaysian Navy has located one body at sea, and officials are working to determine whether it is one of the missing sailors, he said. Divers have located others.
US military divers continue to search the flooded areas of the McCain. Ships and planes are scouring the seas east of Singapore. Divers will also assess the extent of damage to the warship, which is docked at a Singapore naval facility, according to the Navy.
"Until we have exhausted any potential of recovering survivors or bodies, the search and rescue efforts will continue," Swift said.
The White House expressed its "great sadness" about the incident. "As the Navy begins the process of recovering our fallen sailors, our thoughts and prayers go out to their families and friends," the statement said.
The guided-missile destroyer and the merchant vessel collided Monday east of the Malacca Strait, the fourth time a US warship has been involved in an accident in Asian waters this year.
"This trend demands more forceful action," Adm. John Richardson, chief of naval operations, said. Swift said the Pacific Fleet will complete the operational pause by next Monday.
Swift also said there will be "a deliberate reset for our ships focused on a number of areas, such as navigation, ship's mechanical systems and bridge resource management." A reset refers to the restoration of unit or equipment capability and readiness.
"It will involve training and assessment by a team of experts," he said.
At least seven vessels from the Malaysian and Singapore navies, plus military aircraft from the USS America, are involved in the search near one of the world's most congested shipping lanes.
A Malaysian official said the sea in the area was "quite rough," with waves up to 1 meter (3-feet) high.
One of the missing seamen was named as Petty Officer Logan Palmer in a Facebook post by US Rep. Rodney Davis of Illinois, who said he was in touch with Palmer's family.
Five sailors were injured in the crash, and a Singaporean air force helicopter airlifted four of them to a Singapore hospital.
What caused the accident?
The McCain suffered a steering failure as the warship was beginning its approach into the Strait of Malacca, causing it to collide with a commercial tanker,
a Navy official told CNN.
The official said it was unclear why the crew couldn't use the ship's backup steering systems to maintain control.
Earlier, another US Navy official told CNN there were indications the destroyer experienced a loss of steering right before the collision, but steering had been regained afterward.
Swift lauded the "damage control efforts of the crew," such as helping injured sailors and fighting to control the flooding and stability of the ship.
"John S. McCain was up and running as an operational ship almost immediately after the collision. It was quite extraordinary," he said.
The McCain has been towed to Singapore's Changi Naval Base. The other ship, the Alnic MC, a 600-foot, Liberian-flagged oil and chemical tanker, was towed to Singapore's eastern anchorage for inspection.
The collision bears a resemblance to an incident in June involving the USS Fitzgerald,
which collided with a container ship off the coast of Japan and led to the deaths of seven US sailors.
In addition, the USS Lake Champlain hit a South Korean fishing boat in May, and the USS Antietam ran aground off the coast of Japan in January.
Lawmakers and defense analysts have warned that longer deployments for ships and less time and money for maintenance and training could be playing a role in the accidents.
A press release on the McCain last week
said the crew had completed 350 maintenance and repair jobs while at sea. At least 100 of those were classified as "depot-level" jobs that would usually be conducted at a shipyard.
"We are not letting this deployment eat away at our material readiness," said Cmdr. Jessie Sanchez, the executive officer of McCain. "We continue to maintain our upkeep, so that when we come back, we are just as good if not better than when we left."
There was no indication the collision was intentional or evidence the ship was the target of a cyberattack, but investigators would not discount these possibilities, Richardson told reporters.
"We are taking a look at all (options) as we did with the (USS) Fitzgerald as well," he said.
What actions are being taken?
The spate of accidents suggests there could be a more systemic issue.
"This is the second major collision in the last three months, and is the latest in a series of major incidents, particularly in the Pacific theater," Richardson said as he announced the operational pause.
The United States has ordered a "comprehensive review," which Defense Secretary James Mattis said "will determine any of the causal factors, to determine what's going on -- both immediate contributors to this incident but also any related factors."
The review will take place over a week in a series of 24-hour periods during which on-board actions, as well as leadership and operational procedures, will be examined.
Speaking at the Osan Air Base in South Korea, Adm. Harry Harris, commander of the US forces in the Pacific, said the pause would not have an effect on the ability "to defend our nation and our allies."
The McCain, named for Sen. John McCain's father and grandfather, both of whom were Navy admirals, is one of 84 US warships equipped with the Aegis missile defense system, which has been touted as a possible counter to any North Korean missile launch.
Swift said he was not concerned that the number of ships out of commission would affect the Navy's ability to respond to threats in the region.
"We have a deep bench," he said. "I've got 140,000 sailors that report for duty in the Pacific Fleet every day. I have over 200 ships assigned to me, 60% of the United States Navy is assigned to the Pacific Fleet, so I am highly confident we can respond to any guidance or direction."
Nevertheless, Swift said, "We are very concerned about the four incidents."
Retired Navy Rear Adm. John Kirby, a CNN diplomatic and military analyst, said the Navy will look at a number of factors to try to understand if there's a systemic problem.
"They'll look at the quality of leadership at all levels, the amount and the quality of training that commanders have been able to get done, shipboard watch-standing procedures and qualifications, and system and equipment readiness," Kirby said.
"I suspect they will also want to consider the degree to which the budget uncertainty of the last few years has likewise affected any of those factors."
Criticism in Chinese press
An opinion piece in the state-run China Daily newspaper criticized the wave of accidents, saying the US Navy was becoming "a hazard in Asian waters."
"It may be hard for people to understand why US warships are unable to avoid other vessels since they are equipped with the world's most sophisticated radar and electronic tracking systems, and aided by crew members on constant watch," said the unbylined piece published in the English-language
Calling the Navy a "dangerous obstacle" and an "increasing hindrance to ships sailing in Asian waters," it said there was "no denying the fact that the increased activities by US warships in Asia-Pacific since Washington initiated its rebalancing to the region are making them a growing risk to commercial shipping."
It added that investigations into the cause of the Fitzgerald collision in June "shed some light on the way US warships tend to sail without observing maritime traffic rules and the sloppiness of their crews."