The police have denied earlier reports that the man was shot, saying that he may have died having “fallen down”.
A doctor at the hospital has told Reuters news agency that the victim did not have any gunshot wounds.
Last week, there were complaints of police brutality as officers were pictured assaulting demonstrators in the capital, Nairobi.
This is the fourth week of protests over the electoral commission.
The opposition Cord alliance wants the current members of the electoral commission to resign ahead of next year’s presidential election.
Cord is complaining that they are biased in favour of the governing Jubilee coalition and want it replaced by a neutral body, the BBC’s Anne Soy reports from Nairobi.
Deputy President William Ruto told the BBC’s Focus on Africa TV that Monday’s protests were illegal, on the basis that police were not given adequate warning.
“Peaceful protests are safeguarded and provided for by the constitution,” he added.
Mr Ruto also said that police who were caught on camera beating protesters last Monday, sparking a public outcry over the alleged brutality, “will be brought to book”.
Those who “jeopardised the rights of other Kenyans will face the full force of the law,” he said.
A police investigation into the officers’ behaviour is ongoing.
Several other places have been caught up in the demonstrations including the coastal city of Mombasa, where police fired tear gas to prevent protesters from gathering.
In Nairobi, there was a heavy police presence to stop people joining the demonstrations.
The opposition says that one of its senators leading protests in the western town of Kakamega has been arrested.
Last week, President Uhuru Kenyatta said that if the opposition wanted changes at the electoral commission, they should use constitutional procedures to achieve this.
He “urged the opposition to exercise mature… politics instead of using illegal means to attract undue attention”, according to a government statement.