In the letter, Tharoor said the relevant authorities be asked to expedite the investigation and come to a rapid conclusion so that the truth emerges at the earliest regarding the death of 52-year-old Sunanda at a luxury hotel here on Friday evening.
"In the midst of my mourning, I have finally had a chance to catch up with the media reports and am horrified to read the reckless speculation rampant there.
"I am appealing to you to issue instructions to the relevant authorities to expedite their investigation and come to a rapid conclusion so that the truth emerges at the earliest," the Minister of State for HRD said.
"I pledge full and unstinting cooperation. Nothing short of the truth will end the indignity to which my wife and I are being subject at a time when all I seek is to be allowed to grieve in private with her and my near and dear ones," Tharoor, 57, said.
AIIMS doctors, who conducted her post mortem, have said Sunanda had died a "sudden, unnatural" death.
A member of the autopsy team has said her body bore "some physical injuries" but it was not clear if they were related to her death.
Sunanda's death came two days after she went public on Twitter alleging that her husband was having an affair with Pakistani journalist Mehr Tarar.
However, the next day, the couple announced that they were happily married and wanted to remain so.