* Vice president quits on polling day
* Constitution backed by 57 percent in first round vote
* End of voting was pushed back by four hours
* Rights groups report some irregularities
By Shaimaa Fayed and Edmund Blair
CAIRO, Dec 22 (Reuters) - Egyptians voted on Saturday in the second round of a referendum expected to approve an Islamist-drafted constitution that lays foundations for transition to democracy but is criticised as divisive by the opposition.
Counting began after voting, extended by four hours as queues formed at polling stations, closed at 11 p.m. (2100 GMT). Last week's first round, which an opposition leader said was marred by "serious violations", returned 57 percent in favour of the constitution, according to unofficial figures.
Analysts expect another "yes" because the second stage includes rural areas with more Islamist sympathisers. Egyptians exhausted by two years of upheaval may also vote in favour.
Islamist backers of President Mohamed Mursi say the constitution is vital to move to democracy, nearly two years after an Arab Spring revolt overthrew authoritarian ruler Hosni Mubarak. It will provide stability for a weak economy, they say.
But the opposition accuses Mursi of pushing through a text that favours Islamists and ignores the rights of Christians, who make up about 10 percent of the population, as well as women.
"I'm voting 'no' because Egypt can't be ruled by one faction," said Karim Nahas, 35, a stockbroker, heading to a polling station in Giza, a province included in the second, decisive round of voting which included parts of greater Cairo.
At another polling station, some voters said they were more interested in ending Egypt's long period of political instability than in the Islamist aspects of the charter.
"We have to extend our hands to Mursi to help fix the country," said Hisham Kamal, an accountant.
Hours before polls closed, Vice President Mahmoud Mekky announced his resignation. He said he wanted to quit last month but stayed on to help Mursi tackle a crisis that blew up when the Islamist leader assumed wide powers.
Mekky, a prominent judge who said he was uncomfortable in politics, disclosed earlier he had not been informed of Mursi's power grab. The timing of his resignation appeared linked to the lack of a vice-presidential post under the draft constitution.
The referendum committee may not declare a result for the two rounds until Monday, after hearing appeals, although unofficial results may come sooner.
As polling opened on Saturday, a coalition of Egyptian rights groups reported a number of alleged irregularities.
They said some polling stations opened late, that Islamists illegally campaigned at some of them, and complained of voter registration irregularities, including listing of a dead person.
The new basic law sets a limit of two four-year presidential terms. It says sharia law principles remain the main source of legislation but adds an article to explain this further. It also says Islamic authorities will be consulted on sharia - a source of concern to Christians and other non-Muslims.
If the constitution passes, there will be parliamentary elections in about two months. If not, an assembly will have to be set up to draft a new one.
After the first round of voting, the opposition said alleged abuses meant the first stage of the referendum should be re-run.
But the committee overseeing the two-stage vote said its investigations showed no major irregularities in voting on Dec. 15, which covered about half of Egypt's 51 million voters.
If the charter is approved, the opposition says it is a recipe for trouble since it will not have received sufficiently broad backing and that it will not have been a fair vote.
"I see more unrest," said Ahmed Said, head of the liberal Free Egyptians Party and a member of the National Salvation Front, an opposition coalition formed after Mursi expanded his powers on Nov. 22 and then pushed the constitution to a vote.
Protesters accused the president of acting like a pharaoh, and he was forced to issue a second decree two weeks ago that amended a provision putting his decisions above legal challenge.
Said cited "serious violations" on the first day of voting, and said anger against Mursi was growing. "People are not going to accept the way they are dealing with the situation."
At least eight people were killed in protests outside the presidential palace in Cairo this month. Islamists and rivals hurled stones at each other on Friday in Alexandria, the second-biggest city. Two buses were torched.
The head of the Muslim Brotherhood, Mursi's power base, said Egypt could move on after the vote.
"After the constitution is settled by the people, the wheels in all areas will turn, even if there are differences here and there," the Brotherhood's supreme guide, Mohamed Badie, said as he went to vote in Beni Suef, south of Cairo.
"After choosing a constitution, all Egyptians will be moving in the same direction," he said.
The vote was split over two days after many judges refused to supervise the ballot.
Islamists, who have won successive ballots since Mubarak's overthrow, albeit by narrowing margins, dismiss charges they are exploiting religion and say the constitution reflects the will of a majority in the country where most people are Muslim.
In a resignation letter, Mekky said "the nature of political work did not suit my professional background as a judge".
A cabinet spokesman denied a state television report that the central bank governor, Farouk El-Okdah, had resigned.