Nehawu has called for the resignation of Helen Zille as Western Cape premier and the leader of the opposition Democratic Alliance.
"She is not fit to hold office," said National Education Health and Allied Workers Union spokesman, Sizwe Pamla, in a statement on Wednesday.
"She must take responsibility for all the damaged property in the Johannesburg CBD, and the blood of innocent people that was spilt yesterday."
On Tuesday, police used teargas and water cannons to disperse Congress of SA Trade Union members who stopped the DA from protesting outside its headquarters.
The DA wanted Cosatu to stop stonewalling a government youth wage subsidy which it believed would create thousands of new jobs for young people.
Rocks and pieces of cement were thrown into the crowd during Zille's speech and a DA member was seriously injured.
Nehawu condemned the violence and damage of property during the march.
"Workers who gathered in Braamfontein to defend the federation were stoned by the [DA] rent-a-crowd and were also shot at by the police," said Pamla.
"The DA, with their bricks and stones, damaged several cars and hurt many people," he alleged.
He urged those who were injured to send their medical bills to Zille.
"They must take full responsibility for the actions of their rented-mob..."
DA spokesman Musi Maimane said Nehawu was "clearly trying to deflect the blame for the violence that Cosatu members directed towards DA members."
He said DA members were participating in a legal march that was sanctioned by the Johannesburg metro police, while Cosatu's gathering was illegal and "intended to disrupt the DA's march".
"The whole of South Africa knows that Nehawu and Cosatu are trying to protect the interests of their members, whilst the DA is fighting for the rights of the unemployed," he said.
He said the youth wage subsidy would create 420,000 jobs.
"Nehawu is simply grandstanding... their comments are nonsensical."