The latest violence erupted after the government moved to shut down Hezbollah's telecoms network and remove the chief of security at Beirut's airport for allegedly sympathising with Hezbollah.
The military said it would use force - if necessary - to restore order, starting from 0600 (0300GMT) on Tuesday.
For the past 16 months, Lebanon has been locked in political stalemate between the ruling coalition and Hezbollah-led opposition over the make-up of the government.
The army has emerged as the only factor preventing a complete collapse, and it is generally agreed that its commander, Gen Michel Suleiman, should be the next president.
Lebanon has not had a president since November, when Emile Lahoud stepped down despite parliament failing to agree on his successor.
Lebanon was plunged into civil war from 1975-90, drawing in Syria and Israel, the two regional powers.
'Source of instability'
Speaking to the BBC's Arabic television channel, he said Lebanon's success was important for peace in the Middle East.
Gunfire on the streets of Tripoli
He said the US was helping the Lebanese army become effective enough to act against Hezbollah's armed wing.
"I don't see how you can have a society with Hezbollah armed up the way they are.
"In this case though, they moved against the Lebanese people, they're not moving against any foreign country, they're moving against the Lebanese people and it should send a signal to everybody that they're a destabilising force."
"The first step of course is to make sure that the Siniora government has got the capacity to respond with a military that's effective," he said.
Hezbollah would be nothing without Iranian backing, he said, adding that Iran was the source of much instability in the Middle East.
The interview took place ahead of Mr Bush's second - and probably last - trip to the Middle East. He arrives in Israel on Wednesday and will also visit Saudi Arabia and Egypt.
In his BBC interview, Mr Bush also said:
That the status quo between the Israelis and the Palestinians was unacceptable;
That he would push the Israelis and Palestinians to reach, before the end of the year, a "description" of a Palestinian state with defined borders that did not "look like Swiss cheese";
And that there were "concrete examples" of North Korean assistance to Syria to build a nuclear reactor on a site which Israel bombed last year.
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