Seven people died at the National Trauma Centre, said Dr Badri Risal, adding there were 58 wounded in the hospital.
"Many of them are in serious condition and appear to have been shot in the head and chest," the doctor said at the country's main hospital in the heart of Kathmandu.
Families waited anxiously outside for news of their relatives while people lined up to donate blood.
Two people died at the Civil Hospital and two more at the KMC hospital, according to doctors there who spoke on condition of anonymity because they are not authorised to comment.
The wounded were being treated at half a dozen hospitals in Kathmandu.
Tens of thousands of protesters took to the streets of Nepal's capital on Monday to vent their anger against a decision by authorities to block most social media platforms including Facebook, X and YouTube, saying the companies had failed to register and submit to government oversight.
Protesters pushed through barbed wire and forced riot police to retreat as they surrounded the parliament building.
Police fired tear gas and water cannon but were outnumbered and sought safety inside the parliament complex.
Police eventually opened fire at the protesters.
The situation remained tense and the government announced a curfew for Monday around parliament, the government secretariat, presidential house and key parts of the city.
"Stop the ban on social media, stop corruption not social media," the crowds chanted, waving the red and blue national flags.
Monday's rally was called the protest of Gen Z, generally referring to people born between 1995 and 2010.
About two dozen social network platforms that are widely used in Nepal were repeatedly given notices to register their companies officially in the country, the government said.
Those which failed to register have been blocked since last week.
TikTok, Viber and three other platforms have registered and operate without interruption.
The move by the authorities came as the government sent a bill for a debate in parliament that wants to ensure that social platforms are "properly managed, responsible and accountable".
It includes asking the companies to appoint a liaison office or point in the country.
The bill has been widely criticised as a tool for censorship and punishing government opponents who voice their protests online.
Rights groups have called it an attempt by the government to curb freedom of expression and violate fundamental rights.
Nepal in 2023 banned video-sharing app TikTok for disrupting "social harmony, goodwill and diffusing indecent materials".
The ban was lifted in 2024 after TikTok's executives pledged to comply with local laws.
They include a ban of pornographic sites that was passed in 2018.
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