Prisoners are shown being badly beaten, one sexually assaulted with a broom.
In an announcement live on national TV, President Mikhail Saakashvili said police would be deployed in all jails.
Relatives of inmates have tried to storm the prison where the alleged abuses took place.
The ruling party has said the video was staged by the opposition to discredit the government ahead of key elections.
For critics, though, this is graphic proof that the authorities’ tough approach to crime has become unaccountable and heavy-handed.
Mikhail Saakashvili demanded a complete overhaul of the prison system.
“Patrol police must immediately enter all prisons,” he said.
“There must be zero tolerance to any violations of human rights, because we are building a civilized and humane country, rather than discipline based on violence.”
The film showing abuse in the Gldani prison No. 8 was broadcast on TV on Tuesday.
It triggered overnight protests in Tbilisi and also in the southern city of Batumi, Georgian media say.
In the capital, anti-government demonstrations were held in various places, including outside the national broadcaster’s headquarters.
Angry protesters have been demanding the resignation of Khatuna Kalmakhelidze, the minister in charge of prisons.
On Wednesday, Khatuna Kalmahelidze said she had decided to step down because her efforts to protect human rights in prisons had “proved insufficient”.
Later, an opposition TV station released more videos which appear to show the abuse of underage inmates in a juvenile detention centre.
It has long been suspected that prisoners in Georgia are mistreated, our correspondent says.
When Mikhail Saakashvili swept to power after the Rose Revolution in 2004, he clamped down heavily on crime.
But his critics say Georgia has now swung to the other extreme, accusing his government of becoming authoritarian.
The abuse videos come as Georgia is preparing for parliamentary elections on 1 October, seen as the biggest test facing the country’s democracy since the revolution.
President Mikhail Saakashvili’s party faces a challenge from Georgia’s richest man, Bidzina Ivanishvili, who has vowed to oust the government from power.
With a reputation for benevolence and philanthropy, Bidzina Ivanishvili is popular among many poorer voters who are struggling in modern Georgia’s neo-liberal economy, correspondents say.
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