The court has begun hearing opposition candidate Viktor Yushchenko's appeal against the official poll result.
Supporters of his rival, Prime Minister Viktor Yanukovych, have said they will hold a referendum on autonomy if Mr Yushchenko is declared winner.
But the court has now said it will not issue any ruling on Monday.
Partition fears
Outgoing President Leonid Kuchma said on Monday he could not accept any division of Ukraine, after talks with regional leaders who have threatened to demand autonomy.
"My position is that we cannot allow the division of Ukraine," Mr Kuchma said in televised remarks.
His warning was echoed by EU foreign policy chief Javier Solana, who said "the unity of Ukraine is fundamental".
Speaking in Brussels after a meeting with EU and Nato ambassadors, he said dialogue must continue in Ukraine and "what we have to avoid is the outbreak of violence".
Nato Secretary-General Jaap de Hoop Scheffer agreed that maintaining Ukraine's territorial integrity was essential.
"The sense of belonging to one nation is very important and on that basis a solution should be found," he said.
Fears of a split in Ukraine intensified after the eastern Donetsk region on Sunday scheduled a referendum on autonomy.
Officials from some other regions also met to consider seeking autonomy in case Mr Yushchenko - a pro-Western liberal - becomes president.
The parliament of Donetsk region will hold an emergency session on Tuesday to consider the plan to hold a referendum on 5 December.
Focus on court
Near the court building in central Kiev, several thousand rival supporters stood together, shouting "Yu-shchen-ko!" and "Yanu-ko-vych!" and waving the orange flags of the opposition and blue-and-white Yanukovych flags.
Last week the court suspended the official results, which saw Mr Yanukovych, who is seen as pro-Russian, declared the winner with 49.46% of the vote to Mr Yushchenko's 46.61%.
Mr Yushchenko's team has submitted thousands of allegations of ballot-rigging to regional courts and one complaint to the Supreme Court.
But the government has also lodged 7,000 complaints of irregularities to regional courts.
Mr Yushchenko has declared himself the rightful winner, alleging massive fraud. His concerns have been backed by international observers.
The Russian government has suggested it might reverse its opposition to a new election. The EU has already called for a re-run.