New U.S. Ambassador to Seoul Harry Harris said there must be "demonstrable moves" from North Korea toward denuclearization before any declaration of the end of the 1950-53 Korean War is made between the two sides.
Earlier Thursday, Trump thanked Kim on Twitter.
In a statement on Thursday, White House spokeswoman Sarah Sanders said that Mr Trump's tweet was referring to a letter that Mr Trump received on 1 August from Mr Kim.
"I know that President Trump is grateful that Chairman Kim has kept his word, and we see today this tangible progress in our efforts to achieve peace on the Korean Peninsula", said Pence, whose father fought in the Korean War.
Vice-President Mike Pence and USA military leaders received the remains in Hawaii during a sombre ceremony on Wednesday.
The U.S. ambassador also expressed skepticism regarding North Korea's dismantling of the Punggye-ri nuclear test site and the missile site at Tongchang-ri.
Kim also sent a letter to Trump in mid-July in which the North Korean leader said he hoped there would be a second meeting between the two. Senior Trump administration officials have urged patience, cautioning that the process of denuclearizing North Korea and removing the threat of its long-range missiles will take time.
Experts say positively identifying the decades-old remains could take anywhere from days to decades. He said during a Tuesday rally in Tampa, Florida, that the US was "doing well" with North Korea and noted the return of detained Americans and Pyongyang's ceasing of nuclear testing or missile tests.
But there are reports North Korea continues to build nuclear missiles.
More than 400 caskets of remains found in North Korea were returned to the United States between the 1990s and 2005, with the bodies of some 330 other Americans also accounted for, according to the DPAA.