The US-based rights group wrote to Singh ahead of Hu's highly anticipated visit expressing concern at the "deterioration of human rights in Tibet."
"We particularly draw your attention to the recent shooting on the China-Nepal border... We ask that you urge President Hu to permit an independent investigation into the incident," the letter, released to the media late on Monday, said.
Video footage last month showed a line of Tibetans walking through the snow before a shot is heard and a person in the group falls to the ground.
China's official media admitted soldiers killed one Tibetan and injured another neat Mount Everest but said they were acting in self-defence. Foreign climbers who witnessed the incident have disputed China's account.
The footage was shown on a Romanian television station.
Around 2,500 Tibetans a year make a dangerous trip across the Himalayas into Nepal, despite the risks posed by Chinese troops and the elements.
Many of the refugees head on to India's Dharamsala town, where Tibetan spiritual leader the Dalai Lama has lived in exile since 1959.
Human Rights Watch also urged Singh to allow Tibetans "free expression and permit peaceful protests of President Hu's visit." "India's decision to restrict the free speech of activists who have stated their intention to protest peacefully ... is a shocking encroachment on basic human rights," it said.
Last week New Delhi asked prominent Tibetan activist Tenzin Tsundue not to leave Dharamsala till Hu concluded his visit on November 23.
Hu arrived in New Delhi late on Monday on a four-day visit, the first by a Chinese president in a decade, aimed at boosting trade ties and build trust between the giant neighbours who are yet to resolve a decades-old border dispute. Hu will hold talks with Singh on Tuesday.
Ahead of Hu's arrival, hundreds of Tibetans protested in New Delhi and in Dharamsala to demand that Hu hold talks over the Himalayan region.
China has ruled Tibet since it sent in the military to "liberate" the Himalayan region in 1950.

Copyright AFP (Agence France-Presse), 2006