Mr Whitacre added: "All involved agree that changes needed to be made."
He assured GM's employees, dealers, suppliers, union partners and customers that it would be business as usual, with an emphasis on "a return to profitability and repaying the American and Canadian taxpayers as soon as possible".
GM entered bankruptcy protection earlier this year, after having been hit by slumping sales. It emerged from bankruptcy in July, with the US government owning a 62% stake in the revamped firm.
In addition, GM received some $60bn (£36bn) in financing from government loans.
Mr Whitacre was brought in as chairman by the Obama administration earlier this year, having previously run telecoms company AT&T.
Analysts say Mr Henderson, as the right-hand man of the previous chief executive, Rick Wagoner, was seen as too much part of the GM old guard.
Ken Elias, from the car consultancy Maryann Keller, said: "The reality is GM truly needs an outsider as a leader that has no attachment."
Reports suggest GM's board was unhappy with the progress of Mr Henderson's attempted turnaround of the company.
David Bitterman, from Huron Consulting, said: "At the end of the day, it seems a little bit inevitable. Obviously, Whitacre and Henderson didn't envision the new GM in the same way."
The White House denied any involvement in the resignation.
"This decision was made by the board of directors alone. The administration was not involved in the decision," a White House spokeswoman said.
Mr Wagoner, who joined GM in 1977 and had headed the company since 2000, was ordered to step down by President Obama.
The BBC's Jorn Madslien says it could take months before the company finds a replacement for Mr Henderson.
GM emerged from bankruptcy protection in July 2009
"Anyone stepping into his shoes will find his pay packet curbed by presidential decree," he said.
In an agreement reached in October with the US government's "compensation tzar", Kenneth Feinberg, Mr Henderson's pay was cut by 25% to $950,000, about half of what he made in 2008.
The news of Mr Henderson's departure came minutes after the end of a GM board meeting to discuss the fate of its Swedish car brand, Saab.
GM said that because the emergence of new potential buyers, it would evaluate bids for carmaker Saab by the end of December.
But the company added that if it did not find a "suitable arrangement", it would then "wind down" Saab.
A deal to buy Saab, which employs 4,500 people, failed last week after Sweden's Koenigsegg Automotive dropped out of a possible takeover. This was the third failed attempt to sell Saab this year.
In November, GM announced that it was calling off plans to sell its other European business, Opel, along with its UK brand, Vauxhall.
GM had agreed to sell Opel and Vauxhall to Canadian car parts company Magna, but changed its mind.
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