BEIRUT (Reuters) - Lebanese banks said on Monday that a bombing outside the headquarters of one of the country's biggest banks had targeted the entire sector, an attack seen as a dangerous escalation of a crisis over a U.S. law targeting the finances of the Shi'ite group Hezbollah.

The bombing outside the main Blom Bank building in Beirut on Sunday night, which caused damage but no fatalities, followed the closure of accounts considered to be linked to the group by banks afraid of being shut out of the international financial system. Noone has claimed responsibility for the attack.

Hezbollah, which has not commented on the bombing, has condemned the central bank for implementing a law it views as a breach of sovereignty. The banks say they have no choice but to implement it.

Blom Bank, one of Lebanon's biggest banks, has closed more accounts than other banks as a result of the legislation, Lebanese officials say.

Hospitals are among social institutions deemed to be linked to Hezbollah that have had their accounts closed, sources familiar with the matter say.

Societies affiliated to Hezbollah run hospitals, schools and social care institutions and the group says the measures are targeting the entire Shi'ite community.

"The Association of Banks confirms that it operates according to the highest levels of professional practice and within the rules prevailing in the international markets," the Association of Banks in Lebanon said in a statement.

"This explosion targets the entire banking sector."

The U.S. Hezbollah International Financing Prevention Act (HIFPA), passed in December, threatens to punish any organization or person providing significant finance to Hezbollah, designated a terrorist organization by U.S. law.

It poses an unprecedented financial challenge to the Iranian-backed group that dominates Lebanon, its influence underpinned by a powerful armed wing that is fighting in Syria in support of President Bashar al-Assad.

POLITICAL COST

Though analysts say Hezbollah itself can operate without bank accounts, the situation is seen as carrying a political cost as it affects its support base in the Shi'ite community.

"They won a military war, they won a political war, but they cannot win a financial war," said one Lebanese banker, discussing the matter on condition of anonymity. "They have to find ways to work in the shadows but the problem is that whoever deals with them will be under scrutiny."

When the law first came into effect, Lebanese banks began closing accounts, including of Hezbollah officials, themselves. The central bank subsequently issued a directive requiring the banks to refer accounts considered linked to Hezbollah to an investigative unit before any measure is taken.

In a statement issued last week, Hezbollah's parliamentary bloc stepped up its attack on central bank governor Riad Salameh for "ambiguous and suspicious" statements, and said Hezbollah's education and health institutions could not be undermined.

In an interview with CNBC last week, Salameh said the central bank was making sure banks had really studied account activity before any decisions are taken.

"The law is being implemented," he said.

Deflecting criticism, Blom Bank said in a statement on Monday it "represented all sections of Lebanese society and its sects" in terms of its account holders, shareholders and employees.

Parliament Speaker Nabih Berri, a Shi'ite politician and Hezbollah ally, said the attack on Blom Bank had targeted "Lebanon firstly, and Hezbollah secondly".

The banking sector is vitally important to Lebanon as a conduit for billions of dollars of annual remittances that keep its economy afloat. The central bank is widely seen as one of the only effective institutions in the weak Lebanese state.

Nabil Boumonsef, a commentator with An-Nahar newspaper, said the bombing meant the crisis had now become a security issue.

"Its danger is that it is connected to the most important, and I can even say the last bastion of protection for Lebanon's social, political and economic stability - the banking sector."

(Reporting by Tom Perry, Samia Nakhoul, Laila Basam; Writing by Tom Perry; Editing by Catherine Evans, Hugh Lawson and Philippa Fletcher)