The doctors said there has been a rise in "institutional threats, violence and vandalism".
A day after fast-paced developments, when talks between them and the West Bengal government failed to take place for the third straight day and Chief Minister Mamata Banerjee offered to resign, protesting junior doctors have written to the President and the Prime Minister, requesting them to intervene and end the impasse.
The doctors have been staging an agitation since a trainee doctor was raped and murdered in Kolkata's RG Kar Medical College and Hospital on August 9.
In their four-page letter to President Droupadi Murmu, which has been copied to Prime Minister Narendra Modi, Vice President Jagdeep Dhankhar and Health Minister JP Nadda, the doctors wrote, "We humbly place the issues before your esteemed excellency, as the head of state, so that our unfortunate colleague who has been the victim of the most despicable crime shall receive justice, and so that we, the healthcare professionals under the West Bengal Health department, may be able to discharge our duties to the public without fear and apprehension."
Stating that there has been a rise in "institutional threats, violence and vandalism" since the protests began, the doctors added, "Your intervention in these trying times will act as a beacon of light to us all, showing us the way ahead out of the darkness that surrounds us."
Towards the end of August, Ms Murmu had said she was dismayed and horrified by the rape and murder of the trainee doctor.
"What is more depressing is that it (the Kolkata rape and murder) was not the only incident of its kind... it was part of a series of crimes against women," she had said, adding, "In the 12 years since Nirbhaya, countless rapes have been forgotten by society. This collective amnesia is obnoxious."
On Thursday, the West Bengal government had invited the protesting doctors for talks again but they did not attend despite reaching the venue because their demand for a live telecast of the meeting had not been met.
After waiting for the doctors for over two hours, Chief Minister Mamata Banerjee said she was willing to step down in the interest of the people and did not want the post.
"I know many in the delegation were interested in talks. But two of three people were giving instructions from outside. We have seen all that. We could see that because it was being recorded by the press, which were standing right behind... They were giving instructions - 'Do not negotiate, do not go to the meeting'," she said.