Russian Foreign Minister Sergei Lavrov has declared the EU's police and justice mission to Kosovo illegal.
Nato has blamed local Serbian leaders for the border trouble
He told reporters that Brussels' "unilateral decision... is in breach of the highest international law".
A 2,000-strong civilian mission will take shape soon. EU special envoy Pieter Feith began work on Wednesday.
Serbian protesters ransacked two border crossings on Tuesday, forcing UN officials to withdraw and prompting Nato troops to seal the frontier.
Crowds of demonstrators used bulldozers and explosives to demolish the border posts at Jarinje and Brnjak, and US, Estonian and French Nato peacekeepers were called in to restore order.
Nato commander Xavier Bout de Marnhac was quoting as blaming local Serbian leaders for the trouble.
"The leaders should think deeply of their responsibility when they trigger this type of demonstration," he said, according to the AFP news agency.
Students in the Serb-dominated town of Mitrovica are organising daily protests at 12.44 pm, referring to UN Security Council resolution 1244 under which Serbia insists it still has sovereignty of Kosovo under international law.
Russia has already objected to Kosovo's declaration of independence, agreeing with Belgrade's interpretation of the UN resolution and raising the prospect of the established world order being destroyed.
Foreign Ministry spokesman Mikhail Kamynin said that by "pursuing the unilateral scenario of solving the Kosovo problem... the European Union encourages separatism in the world".
Serbia's Foreign Minister Vuk Jeremic was due to convey Belgrade's position to Euro MPs in Strasbourg on Wednesday.
But the EU's special envoy to Kosovo began his task, a day after EU foreign policy chief Javier Solana became the first international statesman to visit Kosovo since independence was declared.
Mr Feith - who will head the international civilian office due to take over from the UN - insisted that the EU mission would be deployed throughout Kosovo despite Serb hostility.
"We need to reach out to the Serb communities here in Kosovo and we are, of course, in touch with Belgrade," he said.
In a separate development, the German cabinet agreed formally to recognise Kosovo's independence on Wednesday. Norway also announced its intention to accept Kosovo as an independent state.
STANCE ON RECOGNITION
For: Norway, Denmark, Finland, Sweden, Germany, Italy, France, UK, Austria, US, Turkey, Albania, Afghanistan
Against: Russia, Spain, Romania, Slovakia, Cyprus
The US, Britain, France, and Italy were among the first to come out in favour of Kosovo's independence but others are opposed.
EU member states set aside differences over the recognition of Kosovo earlier this week by stressing that it was not a precedent for separatists elsewhere.
Spain and several other member states have withheld recognition because of concerns about separatist movements within their own borders.
The UN Security Council is divided over how to respond to Kosovo's move, and it has failed to agree on any action.
China has expressed its deep concern and official media in Beijing announced that a further small deployment of UN peacekeepers would be sent to Pristina next month.
Serbian security forces were driven out of Kosovo in 1999 after a Nato bombing campaign aimed at halting the violent repression of ethnic Albanian separatists.
The province has been under UN administration and Nato protection since then.