(Adds Maduro, Lopez tweets, details of fatalities)
By Andrew Cawthorne and Daniel Wallis
CARACAS, Feb 13 (Reuters) - A Venezuelan court ordered the arrest on Thursday of opposition leader Leopoldo Lopez on charges including murder and terrorism linked to street protests that resulted in the deaths of three people the day before.
Using a slogan "The Exit," the U.S.-educated Lopez has for two weeks helped organize sporadic demonstrations around the country to denounce President Nicolas Maduro for failing to control inflation, crime and product shortages.
The president accuses him of sowing violence to try to stage a coup similar to the one 12 years ago that briefly ousted late socialist leader Hugo Chavez, though there is little indication that the protests could topple Maduro.
"There you have the face of fascism!" Maduro said in a speech to the nation on Thursday night, showing photos and video of Lopez at Wednesday's protest in the capital Caracas, some of the footage set to doom-laden music.
"I tell these fugitives from justice: give yourselves up! ... They should go behind bars," Maduro thundered, saying both the intellectual authors of the violence and those who fired shots had been identified by authorities.
Despite the government's strong words and a brief visit by police to the headquarters of his Popular Will political party, Lopez, 42, was not arrested on Thursday.
Colleagues said he spent the day with advisers at his home in the same wealthy eastern district of Caracas where he was once mayor. After Maduro's late-night speech, he took to Twitter and challenged the president to have him arrested.
"Thanks for all your shows of solidarity. I'm fine. I'm still in Venezuela and I'll stay in the streets. Strength!" Lopez tweeted.
"@NicolasMaduro: don't you have the guts to arrest me? Or are you waiting for orders from Havana? I tell you, the truth is on our side."
Speaking to Reuters on Wednesday, shortly before a court upheld a request from the Public Prosecutor's Office to order his detention, the opposition leader blamed armed government supporters for firing on peaceful protesters.
"The government is playing the violence card, and not for the first time. They're blaming me without any proof," he said.
"NO MORE BLOOD"
While many Caracas residents stayed home on Thursday, there were sporadic student protests around the city.
Some groups of demonstrators blocked streets and burned tires. Bands of motorcyclists roamed the streets. And opposition supporters in the wealthier suburbs east of the capital banged pots and bans from windows in a traditional form of anti-government protest in some parts of Latin America.
"We want solutions to problems, not endless confrontation and violence," said student Manuel Armas, 19, outside the Alejandro Humboldt University, where around 200 protesters waved banners saying "No More Blood."
Scores of government supporters gathered outside the Public Prosecutor's Office building that was vandalized on Wednesday, chanting pro-Maduro slogans and denouncing "fascist violence."
Coming almost a year after the death of Chavez, the unrest has been the latest demonstration of the OPEC nation's polarization and the deep mistrust between both political camps.
Maduro, a 51-year-old former bus driver and union activist who has staked his presidency on maintaining Chavez's leftist legacy, said further protests would not be allowed. Government supporters would march in Caracas on Saturday, he said.
Wednesday's fatalities were Juan "Juancho" Montoya, a community activist from a militantly pro-government neighborhood in the poor west end of Caracas; Neyder Arellano, a pro-opposition student; and Bassil Dacosta, who was identified by fellow protesters as a student but by Maduro as a carpenter.
Some 66 people were injured and 70 arrested after Wednesday's violence, officials said. Some protesters, many with their faces covered, threw stones and lit fires in the streets.
Bolivia, Cuba and Argentina, three of Venezuela's fellow leftist political allies in the region, sent messages of support to Maduro's administration.
"Cuba condemns the coup intentions ... organized by fascist groups," the statement from Havana read.
The protests have exposed rifts within the opposition leadership, with some favoring a more moderate approach and saying marches that turn violent only play into the government's hands as it accuses them of being "saboteurs."
Sporadic political protests have become common over the last decade, but they usually fizzle out within days as residents grow tired of blocked streets and the smell of burning tires.
Opposition moderates note that their biggest successes, such as turning pro-Chavez strongholds into opposition territory, came from leaders stepping away from theatrical street protests to focus on daily issues for voters such as poor services, widespread corruption and one of the world's worst murder rates. (Additional reporting by Caracas bureau reporters, Javier Lopez in Tachira, Daniel Ramos in La Paz, Daniel Trotta in Havana; Editing by Kieran Murray and Meredith Mazzilli)