The protests rocked the start of President Yoweri Museveni’s fourth term in office and prompted a fierce reaction from the security forces and a crackdown on the opposition.
“The known circumstances of the deaths to date indicate that the use of lethal force was unnecessary and unjustified,” said a statement released by the groups, which include 95 Ugandan organisations and New-York based Human Rights Watch (HRW).
“The government has arrested hundreds of protesters and put significant resources into investigating alleged acts of looting, arson and destruction of property, but little effort has gone into investigating killings by security forces,” they said.
The groups called on Museveni to invite the United Nations Special Rapporteur on extrajudicial, summary or arbitrary executions to visit Uganda to investigate the deaths.
A police spokeswoman told Reuters that internal investigations were still in progress.
Some government officials have said those killed were violent protesters. HRW said three had been shot in the back while trying to flee the violence.
One policeman has been arrested for the shooting dead of a two-year-old girl during a separate day of protests but no other security personnel have been held.
Museveni accused the police and judiciary last month of being too soft on the demonstrators and said opposition leader Kizza Besigye was using drug users to spark unrest.
He called teargas a “good solution” for protesters during his annual state of the union speech on June 2.