Mr Turchynov was appointed following the dismissal of President Viktor Yanukovych by MPs on Saturday.
Mr Yanukovych's rejection of an EU-Ukraine trade pact triggered the protests that toppled him.
Russia, which had backed Mr Yanukovych, has recalled its ambassador to Ukraine for consultations.
Earlier Russian Foreign Minister Sergei Lavrov said the opposition "had in effect seized power in Kiev, refused to disarm and continued to place its bets on violence".
The US has said parliament's actions were legitimate and has warned Russia against military intervention.
Mr Turchynov, a close ally of former Prime Minister Yulia Tymoshenko, gave a televised address late on Sunday, hours after his appointment as interim president.
He vowed to set up a "government of the people", and said: "We have to return to the family of European countries."
He added: "We are ready for a dialogue with Russia... on a new, fair, equal and neighbourly basis, acknowledging and taking into account Ukraine's European choice."
European Union foreign policy chief Catherine Ashton has announced she is visiting Kiev on Monday to discuss EU support "for a lasting solution to the political crisis and measures to stabilise the economic situation".
Mr Turchynov said MPs had until Tuesday to form a new unity government.
Thousands of opposition supporters remain in Kiev's Independence Square, heeding opposition calls not to disperse.