In a TV address, he said Moscow wanted the new interim government to react to provocations so it could annex Crimea.
Russia's UN ambassador said any troop movements in Crimea were within an existing arrangement with Ukraine.
US President Barack Obama warned of the "costs" of any Russian intervention in the Ukraine.
President Turchynov appealed to Russian President Vladimir Putin to "stop provocations and start negotiations".
He said Russia was behaving as it did before sending troops into Georgia in 2008 over the breakaway regions of Abkhazia and South Ossetia, which have large ethnic Russian populations.
"They are implementing the scenario like the one carried out in Abkhazia, when after provoking a conflict, they started an annexation of the territory," President Turchynov said.
Protecting 'strategic sites'
His statement came a few hours after the Kremlin said President Putin had spoken of the "extreme importance of not allowing a further escalation of violence" during telephone conversations with Western leaders.
However, flights from and to the Crimean capital, Simferopol, were cancelled with airlines saying airspace over the peninsula had been closed.
Senior Ukrainian official Sergiy Kunitsyn told local media 13 Russian aircraft carrying nearly 2,000 suspected troops had landed at a military air base near Simferopol.
This remains unconfirmed.
Russia's ambassador to the UN, Vitaly Churkin, said that any Russian military movements in Crimea were within Moscow's long-standing arrangement with Ukraine on the deployment of military assets.
"We are acting within the framework of that agreement," he said, after a closed-door meeting of the UN Security Council.
He did not give details of any Russian military deployment.
Earlier in the day, Russian armoured vehicles and helicopters were seen in and around Simferopol and Sevastopol, where Russia's Black Sea Fleet are based.
Armed men in unidentified military uniforms, believed to be Russian soldiers, have moved in on Crimea's parliament, state television building and telecommunication centers.
They are also patrolling the airports in Simferopol and Sevastopol.
The editor of the main television station told the BBC he was informed the men were protecting strategic points in the area, in this case the broadcasting tower.
Ukraine has formally lodged a protest with Russia over what it called a violation of Ukrainian airspace and a failure to observe an agreement on the stationing of Russia troops in Crimea.
On Friday, Ukraine's former President Viktor Yanukovych made his first public appearance since being ousted from office a week ago.
At a news conference in Russia, he apologised for not "having enough strength to keep stability" in Ukraine and called his usurpers "young, neo-fascist thugs".
Mr Yanukovych said he would "continue to struggle for the future of Ukraine", but said he would only return if his safety could be guaranteed.
Ukraine has started procedures demanding his extradition.
He is wanted on suspicion of mass murder following violent clashes between police and protesters last week that left more than 80 dead.
Kiev had been in political deadlock since last November when Mr Yanukovych rejected a trade deal with the EU in favour of a similar agreement with Russia.
The move brought thousands of Western-leaning protesters out on to the streets calling for his resignation and closer ties with the EU.
Since he was deposed, the tensions have shifted to Crimea, where the majority of the population are ethnic Russians.
From the scene
Passengers at the main airport in Simferopol were waiting for their flight to Istanbul when they were told airspace over the city was closed and the flight would not leave until the next morning at the earliest.
Some considered making the 500km (310-mile) trip to the nearest international airport, in Odessa.
Meanwhile, the airport car park was still being patrolled by heavily-armed soldiers in uniforms that gave no indication of where they were from.
News from elsewhere was no less alarming.
The main television station was taken over by armed men who said they were from Russia's Black Sea Fleet.
Reports came in that fibre optic lines connecting Crimea to the rest of Ukraine were either blocked or damaged.
If the mobile or internet connection goes down, it will be the most palpable indication yet of an emergency situation for the local people.
Source: BBC News