Four cars in the Indian president's convoy crashed as they approached a roundabout in the university city of Uppsala.
Crown Princess Victoria and Prince Daniel were also making the journey but escaped unhurt, according to the Swedish Royal Court.
But several others were taken to hospital following the accident, which took place just after 4.30pm.
The number of people injured was originally reported as six, but police later amended the figure to nine.
“I have information saying nine people. Two of them are supposed to have been put in an ambulance, the rest were taken to hospital in police cars. No one is seriously injured. But this information could change,” Christer Nordström, press officer at Uppsala police, told Swedish tabloid Aftonbladet.
By 8.30pm on Tuesday all except three who were still waiting to be X-rayed had left hospital.
Swedish police are investigating the incident as a workplace accident.
IN PICTURES: India's President in Sweden
The Swedish royals and President Pranab Mukherjee were making their way to Uppsala University, where India's head of state was due to give a speech as part of a three-day visit to the Nordic nation.
He continued his journey to the Swedish institution, where he was joined by the royal couple, and left the country as planned on Tuesday evening.
— President of India (@RashtrapatiBhvn) June 3, 2015
President Pranab Mukherjee is the first Indian head of state in history to visit Sweden.
He arrived over the weekend at the invitation of King Carl XVI Gustaf and was welcomed at Stockholm's Arlanda Airport by Crown Princess Victoria and Prince Daniel before being given a ride in a horse-drawn cortege through the streets of the Swedish capital.
He has also held meetings with business leaders and the Swedish Prime Minister, Stefan Löfven.