He also confirmed that the Bank has been making contingency plans for the break-up of the euro.
The rate of inflation will remain above the government's 2% target "for the next year or so", the Bank said.
Sir Mervyn was presenting the Bank's quarterly inflation report.
He told a news conference that the euro area posed the greatest threat to the UK recovery, and there was a "risk of a storm heading our way from the continent".
"We have been through a big global financial crisis, the biggest downturn in world output since the 1930s, the biggest banking crisis in this country's history, the biggest fiscal deficit in our peacetime history, and our biggest trading partner, the euro area, is tearing itself apart without any obvious solution.
"The idea that we could reasonably hope to sail serenely through this with growth close to the long-run average and inflation at 2% strikes me as wholly unrealistic," Sir Mervyn said.
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European policymakers, I suspect, will not rush to thank him for his kind and timely advice”
Andrew Balls, the managing director in London of global investment firm Pimco, said it was reasonable for Sir Mervyn and other policymakers to plan for a Greek exit.
"If Greece is to slide out of the euro and collapse, how are they going to protect Ireland, Portugal, Spain and Italy?"
Separately, Prime Minister David Cameron also spoke of the financial storm clouds across Europe, warning that eurozone leaders must act swiftly to solve its debt crisis or face the consequences of a potential break up.
He said during Prime Minister's Questions in the House of Commons: "The eurozone has to make a choice. If the eurozone wants to continue as it is, then it has got to build a proper firewall, it has got to take steps to secure the weakest members of the eurozone, or it's going to have to work out it has to go in a different direction,
"It either has to make up or it is looking at a potential break up. That is the choice they have to make, and it is a choice they cannot long put off."
The Bank's report said, however, that the eurozone crisis was not the only issue weighing on the UK economy, with volatile energy and commodity costs, and the squeeze on household earnings also having an impact.
Andrew Balls, of global investment firm Pimco says, "a disorderly outcome for Greece is going to be bad for the global economy".
Nevertheless, he remained optimistic about the longer term. "We don't know when the storm clouds will move away. But there are good reasons to believe that growth will recover and inflation will fall back," he said.
On quantitative easing, he said that no decisions had been made whether or not to continue pumping money into the economy. The last stimulus programme was still "working its way through the system".
'Outlook is probably better'
Sir Mervyn's comments came on the day that official unemployment figures showed a fall in the jobless rate, underlining recent surveys that the private sector had become more confident about hiring labour.
He said the fall in joblessness was consistent with the expected gradual recovery in the UK economy.
But Graeme Leach, chief economist at the Institute of Directors, said of the Bank's report: "Talk about kicking an economy when it's down.
"On top of the euro crisis and a double-dip recession, the Bank of England is now saying inflation may not fall fast enough to permit more quantitative easing.
"Actually we think the inflation outlook is probably better than the Monetary Policy Committee (MPC) thinks, with the impact of the euro crisis, declining real incomes and weak money supply growth suggesting inflationary pressures may recede later this year and into 2013.
"After many years of underestimating inflationary pressure let's hope the MPC is now making the opposite mistake by overestimating it".
Ed Balls, Labour's shadow chancellor, said: "The Bank of England has once again slashed its growth forecast for Britain, but despite this the government says it will just plough on regardless with policies that are hurting but not working.
"The governor is right to warn of a coming storm from Europe. That is why we warned George Osborne not to rip up the foundations of the house and choke off Britain's recovery with spending cuts and tax rises that go too far and too fast.
"What happens in the eurozone in the coming weeks and months will have an impact on our weakened economy," Mr Balls added.- BBC
Eurozone was 'very close to collapse'
A European Central Bank board member has conceded the ECB may have "saved" the eurozone banking system and eurozone economy in Autumn 2011 by providing one trillion euros of emergency loans to hundreds of European banks at an interest rate of just 1%.
ECB Executive Board member, Benoit Coeure, told Robert Peston: "We were very close to a collapse in the banking system in the euro area, which in itself would have also led to a collapse in the economy and deflation, And this is something that the ECB could not accept."
