The UH-1Y Huey had been delivering aid when it disappeared on Tuesday, with the wreckage only found on Friday after a search mission involving helicopters and hundreds of American and Nepalese troops.
"Today our hearts are heavy with grief for the US Marines who perished when their helicopter went down in the mountains of Nepal earlier this week while providing aid to earthquake victims there," Carter said in a statement.
"We also join our Nepalese partners in mourning the loss of their service members who were onboard the helicopter at the time."
The US defence chief added: "Our mission continues in Nepal, and we remain dedicated to answering the call when disaster strikes, both in the Asia-Pacific and around the world."
John Wissler, the commander of the joint task force investigating the helicopter's disappearance earlier confirmed the wreckage of the helicopter had been found, describing the incident as a "severe crash".
He said he could not yet "positively identify the cause of the mishap", adding that recovery operations would resume on Saturday after being called off due to heavy thunderstorms.
The aircraft was spotted in a remote forest around 70 kilometres (40 miles) northeast of Kathmandu, according to Nepalese army official Major General Binoj Basnet.
Basnet told AFP that troops had reached the crash site on foot and two choppers, including a US military helicopter, had managed to land in the mountainous region after strong winds had kept them circling the wreckage.
Army helicopters and hundreds of US and Nepalese ground troops had been deployed to scour the mountainous terrain where the US chopper disappeared on the same day that a second major earthquake hit the country.
Before it went missing, there was "some chatter about a fuel problem" on the radio from the helicopter crew, Pentagon spokesperson Colonel Steven Warren said on Tuesday.
Relief teams from around the world have been working for weeks to provide water, food, shelter and medical assistance to Nepal after the first, 7.8-magnitude quake hit on April 25.
Nearly 8,500 people have now been confirmed dead in the disaster, which destroyed more than half a million homes and left huge numbers of people without shelter with just weeks to go until the monsoon rains.
The missing helicopter was among more than a dozen US military aircraft devoted to aid operations, including two other Huey choppers and four tilt-rotor Ospreys as well as cargo planes.
The United Nations has said it faces a "monumental challenge" to bring relief to victims, many of whom live in areas accessible only on foot or by helicopter.
Jamie McGoldrick, UN humanitarian coordinator for Nepal, on Friday warned of "more deaths" to come in quake-hit regions unless donors dramatically ramp up aid contributions, which currently stand at just 14 percent of the $423 million appeal launched on 29 April.
"If we don't act quickly, the implications will be severe," McGoldrick said in a statement.
"We can only expect misery, a crippling loss of dignity and the real potential for more deaths especially in the rural and remote areas."
British aid helicopters turned back from Nepal
At least three British helicopters were sent to help with rescue efforts in Nepal without being used, the British government.
"We are disappointed that our Chinooks will not be supporting the World Food Programme's request for help in distributing aid," a government spokesman said.
The three Royal Air Force Chinook helicopters had not reached Nepal but were stranded in Delhi, India, for a week after being sent to bring supplies and rescue stranded victims after the first quake on April 25.
"All decisions on the relief effort are ultimately for the government of Nepal and they have advised that there is no need for the Chinooks to remain forward-based in Delhi," the spokesperson said.
The Nepalese government has said it was overwhelmed by the scale of the first earthquake, the deadliest to hit the country in over eight decades.
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