49-year-old Tembe, a naval aviator, had assumed command of Truman only three months ago. Mumbai-born Tembe came to the US as a child and later graduated from Texas Tech University.
In a statement, the US Navy said, the ship's medical response team provided immediate medical assistance until Tembe could be transported to Bons Secours Maryview Medical Centre where he was later pronounced dead.
The cause of death has not been determined, it said.
"We offer our sincere condolences to Capt Tembe's wife and children, his family and the Truman crew. They are in our thoughts and our prayers as we deal with this tragic loss. Capt Tembe served the Navy and our nation honourably and with great distinction. We honour his outstanding contributions to our nation," said Rear Adm Ted Branch, commander, Naval Air Force Atlantic.
Tembe is survived by his wife and two sons. Planning for a memorial service and funeral arrangements is ongoing.
The ship's executive officer, Capt Craig Clapperton, has temporarily assumed command.
Capt Clapperton described Tembe as an exceptionally strong leader and motivator who went to great lengths to connect with his sailors and understand their challenges.
"He earned their trust and their confidence," Clapperton was quoted as saying by Virginian Pilot newspaper.
Many on the ship regard Tembe as the best commanding officer they've ever had, he added.
Clapperton said the cause of Tembe's death still has not been determined, and that he had no known medical problems.
Mumbai-born Tembe came to the US as a child with his family. He grew up in New York City where, he told the newspaper in an earlier interview, he attended school in Brooklyn and loved the cultural melting pot of his neighbourhood.
Tembe was trained as an EA-3B pilot and served also as an instructor with the Strike Aircraft Test Directorate, Patuxent River Naval Air Test Centre. Other assignments included Strike Fighter Squadron (VFA) 132 on the USS Forrestal and VFA-83 on the USS Enterprise. He also served as executive officer of the USS Dwight D Eisenhower.
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