"We feel a high-level independent inquiry needs to be set up to look into the incident. The inquiry needs to go to the bottom of it and punish responsible people, however high they may be," PIPFPD India chapter said in a statement.
PIPFPD demanded that once Sarabjit is better, he should be released and repatriated to India.
Singh was attacked Friday last at Lahore's Kot Lakhpat Jail, with bricks and plates. He is reported to be critical, and in coma, in a local hospital.
PIPFPD general secretary E. Deenadayalan and joint secretary Jatin Desai said Singh had apprehended attacks on him, and even his sister Dalbir Kaur had expressed her concerns earlier.
Pakistani authorities have initiated a probe into the incident and registered complaints against two inmates, both death row convicts.
Charged with involvement in serial blasts in Lahore, Multan and Faisalabad, and causing at least 14 deaths, Singh was awarded the death sentence and has been in the Pakistan jail over over 21 years.
His mercy plea was rejected by former Pakistan President Parvez Musharraf, but when the Pakistan Peoples Party came to power in 2008, it stayed Singh's execution for an indefinite period.
In the meantime, many peace-loving Indian and Pakistani people and groups around the world have repeatedly appealed to Pakistan to release and repatriate Singh to his home country.
Both Deenadayalan and Desai urged both India and Pakistan to take care of each other's prisoners and ensure that similar incidents do not happen in future.
PIPFPD also urged India and Pakistan to consider a plan under which any person awarded long years of sentence can be transfer to his/her own country, to serve the remaining sentence in their own country.
India has such an agreement with Bangladesh, PIPFPD pointed out.