Greece's finance minister has acknowledged the concerns
Falling confidence in Greece's economic policies and ability to pay its debts have seen Greek shares slump 6%.
The decline came after the Fitch credit rating agency cut Greece's rating to a 10-year low. Greek bond prices also fell as a result.
Fitch's move underscores the fact that Greece's debt is the most expensive to service in the eurozone.
The European Commission urged the Greek government to take "more measures" to reduce its deficit.
Greece has the highest debt of the 16-member eurozone - forecast at 125% of gross domestic product next year.
Fitch's downgrade follows recent warnings by two other agencies that they were considering such a move.
"A difficult situation in one euro area member state is a matter of common concern for the euro area as a whole," warned outgoing European Union economic and monetary affairs commissioner Joaquin Almunia in a statement.
"It is clear that Greece faces very substantial economic and fiscal challenges... but more measures are required."
On Monday, Jean-Claude Trichet, the president of the European Central Bank (ECB), said the Greek government needed to take "courageous" measures.
Greece's Finance Minister George Papaconstantinou has acknowledged growing fears about its ability to pay its debts, fuelled by Dubai's financial problems.
Some analysts say Greece could at some point be unable to use its own government debt as collateral for borrowing from the ECB.
"Greece finds itself in an asphyxiating stranglehold to prod the government to undertake structural reforms and measures to combat tax evasion," said Manos Hatzidakis, an analyst with brokerage firm Pigasos.
The BBC's economics editor Stephanie Flanders said that although there are no formal bail-outs for eurozone economies, Greece has been getting the next best thing to a bail-out, from the ECB.
The financial blow came as the government struggled to settle two days of youth riots.
They mark the anniversary of the police shooting of a 15-year-old boy last year. That triggered widespread violence also fuelled by anger over the economy.
Governments and companies issue bonds to raise money on the financial markets.
Markets are worried about the vulnerability of government bonds following comments from Dubai's government minister that it would not guarantee troubled property firm Dubai World's debt.
Other countries seen as weak include the Irish Republic, Spain and Portugal.