By David Brunnstrom and Timothy Gardner

WASHINGTON (Reuters) - U.S. President Barack Obama met the exiled Tibetan spiritual leader, the Dalai Lama, at the White House on Wednesday despite a warning by China that it would damage diplomatic relations.

The meeting came at a time of heightened tensions between the United States and China over Beijing’s assertive pursuit of territorial claims in East Asia.

Obama's fourth White House meeting with the Dalai Lama in the past eight years took place in the residence of the executive mansion, instead of the Oval Office where the president normally meets world leaders.

White House spokesman Josh Earnest said the choice of the residence emphasized the "personal nature of their meeting."

Obama urged "meaningful and direct dialogue" between the Dalai Lama and his representatives with Chinese authorities to lower tensions and resolve differences, the White House said in a release.

China considers the Dalai Lama a dangerous separatist, and Foreign Ministry spokesman Lu Kang said the meeting would encourage "separatist forces." He urged Washington to abide by its promises to recognize that Tibet is part of China and cease any support for Tibet independence.

Obama told the spiritual leader he does not support Tibetan independence, the White House said.

The president thanked the Dalai Lama for his condolences for the victims of Sunday's mass shooting at a gay club in Orlando, Florida, Earnest said.

Obama had in the past spoken of his "warm personal feelings" for the Dalai Lama, appreciation of his teachings, and belief "in preserving Tibet's unique religious, cultural and linguistic traditions," Earnest added.

China's Foreign Ministry said earlier it had lodged diplomatic representations with the United States over the planned meeting, saying it would damage Chinese-U.S. ties.

A commentary on China's official news agency Xinhua accused Washington of breaking its promise not to support Tibet's independence by going ahead with the meeting. It said this had "seriously jeopardized China-U.S. relations, and deeply hurt the Chinese people's feelings."

When Obama last met the Dalai Lama at the White House in 2014 he angered China by vowing "strong support" for Tibetans' human rights. On Wednesday, Obama and the Dalai Lama also discussed "the equal protection of human rights of Tibetans in China," the White House said.

The Dalai Lama, who fled Tibet into exile in India in 1959 after a failed uprising against Chinese rule, says he wants genuine autonomy for Tibet, not independence.

In an interview with Reuters on Monday, the Dalai Lama said the disputes between China and its neighbors in the South China Sea should be resolved through dialogue.

He said China faced no threat from other Asian countries and as a big and ancient nation should pursue reconciliation and friendship. “Long term, it’s in China’s own interests,” he said. “Trust and friendship with neighboring countries is essential; including the United States also.”

On Wednesday, a Chinese observation ship shadowed the U.S. aircraft carrier John C. Stennis in the Western Pacific, the carrier's commander said, as it joined warships from Japan and India for drills close to waters Beijing considers its backyard.

Japan and the United States worry that China is extending its influence into the Western Pacific with submarines and surface vessels as it pushes territorial claims in the neighboring South China Sea, expanding and building on islands.

China has been angered by what it views as provocative U.S. military patrols close to the islands. The United States says the patrols are to protect freedom of navigation.

On Tuesday, Beijing also warned Washington to stick to its “one-China” policy over Taiwan ahead of an overseas trip by Taipei’s new president, Tsai Ing-wen, who will transit though Miami from June 24-25 and Los Angeles from June 30-July 1 on her way to and from Panama and Paraguay.

A U.S. State Department spokeswoman, Anna Richey-Allen, said the visit would be “private and unofficial” and the transits handled in accordance with the “unofficial nature of the (U.S.) relationship with Taiwan.”

“There is no change to the U.S. ‘One-China’ policy,” she said, adding that Tsai would be greeted as a courtesy in Miami and Los Angeles by officials of the American Institute in Taiwan, a de facto embassy that handles U.S. relations with Taiwan.

(Reporting by David Brunnstrom; Editing by Chizu Nomiyama and James Dalgleish)