The Buddhist-Muslim enclave lost its autonomy in 2019 when it was carved out of Jammu and Kashmir state by Prime Minister Narendra Modi's government and placed under the direct administration of officials in New Delhi.
Protesters, led by activist Sonam Wangchuk, also want Ladakh to be given special status that would allow the creation of elected local bodies to protect its tribal areas.
Late on Wednesday, India's ministry of home affairs said that the protests were triggered by "provocative" speeches by Wangchuk, who had been on hunger strike since September 10.
A mob attacked a political party office, set fire to the Leh chief executive councillor's office and torched a police vehicle, the ministry said in a statement.
More than 30 police and personnel were injured during the incident, it said.
"In self-defence, police had to resort to firing in which unfortunately some casualties are reported," the statement added.
The ministry added that the situation had been "brought under control by 4pm, barring the unfortunate incidents earlier in the day".
"During this (Wednesday's) violence, 2-3 of our youth have died for our cause ... I want to reassure the people of Ladakh that we will not let the sacrifices of these youth go to waste ... We will keep trying to get our demands fulfilled," Thupstan Tswang, chairman of Leh Apex Body, said.
The office of Modi's Bharatiya Janata Party in the main city of Leh was among the buildings vandalised and set on fire, news agency ANI reported.
ANI visuals showed black smoke emanating from behind the office complex's boundary wall and elsewhere, hundreds of people chanted slogans.
Indian TV channels showed an abandoned police vehicle with flames emanating from its front.
Local media reports said some young protesters pelted stones at police and tear gas was used to disperse them.
More than 50 people, including 20 police personnel, were injured, a police source said.
"It was the frustration of the youth ... that brought them to the streets ... I appeal to the youth ... don't walk on this path of violence," said Wangchuk, who called off his fortnight-long hunger strike after the violence.
"This is not a solution to Ladakh's problem ... If our youth have sorrow and pain that we are on hunger strike, then we are breaking our hunger strike from today."
Kavinder Gupta, Ladakh's lieutenant governor, appealed for an end to violence and restoration of peace in a video message circulated by his office.
Demonstrations, public gatherings and inflammatory speech were being banned to maintain peace, district administrator Romil Singh Donk said in a public notice.
Ladakh shares a border with China and Pakistan, and is a strategic territory for India.
India's home ministry has been in talks with Ladakh's leaders since 2023 and has said it is looking into their demands.
The next round of discussions is scheduled for October 6.
