The violence broke out on Thursday as more than 20,000 fans pushed to gain entry to a stadium in Bangalore where the former star's body was on display.
Trouble continued as the star's body was taken to the grounds of a film studio for burial.
Fans torched a police vehicle and battled officers who fired tear gas to keep them at bay.
Gopal Hosur, the joint commissioner of police, said a policeman had opened fire on the crowd "in self-defence", killing one man.
Inside the stadium women, children and elderly men carried flowers, garlands and incense sticks to heap over the transparent funeral casket.
They shouted slogans such as "Long Live Rajkumar" while many wept and beat their chests.
"By such acts you will only be showing disrespect to him"
Karnataka Chief Minister
The casket was then taken in an open truck to the studio where Rajkumar shot most of his films, in a procession that snaked its way through narrow roads packed with the 77-year-old former actor's fans.
State authorities had earlier appealed for calm in the city, where mourning fans had torched buses and cars overnight after the star died of a cardiac arrest.
"By such acts you will only be showing disrespect to him. I request you not to bring discredit to his name," said HD Kumaraswamy, chief minister of Karnataka state.
Shops, business establishments, banks, government offices and technology companies remained closed for two days of mourning.
State authorities had earlier
appealed for calm
The Indian flag flew at half-mast on all government buildings.
Rajkumar starred in over 200 films, playing roles ranging from tough cop to spurned lover.
Rajkumar, whose real name was Mutturaju Singanalluru Puttaswamayya, hit the headlines when the most-wanted Indian bandit, Veerappan, stormed into the film star's farmhouse and abducted him in July 2000.
He was freed after 109 days.
The kidnapping by Veerappan, who was shot dead by police in 2004, gripped the nation and brought rioting fans onto the streets in protest.