The credit rating agency Moody's has downgraded 16 Spanish banks along with Santander's UK arm, citing the Spanish government's reduced ability to shore up the banks.

Santander UK, whose rating was lowered to A2 – a notch higher than its Spanish parent – insisted there was no impact on its business and that it was an autonomous subsidiary with 90% of its assets held in the UK, where it is also regulated. Moody's admitted it was unlikely that the UK Financial Services Authority would allow Santander UK to substantially weaken itself in order to support the parent. increasing instability.

Earlier Moody's had also downgraded four regions in Spain.

The moves from the credit rating agency came after a day of fears that Spain will be the next domino to fall if Greece leaves the euro, as the country was forced to deny that there was a run on its fourth-biggest lender, Bankia.

Bankia shares fell almost 30% at one point after reports – later refuted – that depositors were withdrawing their funds. It was bailed out only last week when the government converted loans into a 45% stake.

The European commission warned last week that high debts in the regions, which account for half of public spending, could prevent Spain meeting its deficit goal of 5.3% of GDP this year.

"It is not true that at this time a flight of deposits from Bankia is taking place," said Fernando Jiménez Latorre, the deputy economy minister. El Mundo newspaper reported that clients had withdrawn €1bn (£800m) of deposits from the bank since last week's bailout.

This appears to have sparked jitters among investors and some clients, though there was no sign of panic in branches. "If they take my money away, I'll kill," one Bankia client spat at budget minister Cristóbal Montoro after approaching him on the street, according to El Mundo.

The bank itself reassured clients in a statement sent to Spain's stock market supervisor. The new chairman, José Ignacío Goirigolzarri, said: "Bankia's clients can be absolutely calm about the security of the savings they have deposited."

Bankia said operations at its branches had "been within normal parameters" during a fortnight of what it called "a highly seasonal nature". It also predicted that "the level of deposits will not suffer any major changes over the coming days".

That was little consolation to 400,000 Spanish investors who were last year persuaded to buy shares in the new bank, which was created by the amalgamation of seven savings banks. The shares have lost 70% of their value since the flotation in the summer.

"It was a shotgun marriage," the head of Madrid's regional government, Esperanza Aguirre, complained, blaming the central bank. The new shareholders have seen their investments slide for 10 straight days and some analysts saw the sell-off as an attempt to cut losses rather than a reaction to rumours of a run on the bank.

Prime minister Mariano Rajoy continues to insist that Spain must pursue austerity; on Wednesday, he called on Europe to take measures, too, warning that Spain risked being frozen out of capital markets as its bond yields crept ever higher, raising the interest rates it must pay on debt.

Spain's borrowing costs shot up further on Thursday with the country's treasury paying about 5% to attract buyers of three- and four-year bonds.

The government took over Bankia, which holds 10% of Spanish deposits, in an attempt to dispel concerns over toxic real estate assets left over from a 2008 property crash.

Spain's economy shrank by 0.3% in the first quarter, putting it back into recession and with a long downturn in prospect as the government cuts spending in an attempt to wrestle down its budget deficit. Unemployment is already nearing 25%, with half of the young without a job.

Anxiety about the eurozone spread to other banks, with shares in Barclays and the bailed-out Royal Bank of Scotland and Lloyds Banking Group all down more than 3% on anxiety about their exposure to the single currency.

UK banks have cut their exposure to Greece to under £6bn. Shares in the French bank BNP were down 4.1%, while Italy's Banca Popolare di Milano and Spain's Bankinter also fell more than 4%.

• This article was amended on Friday 18 May 2012 to make clear that Santander UK has been downgraded to a notch above its Spanish parent company.