* New York governor says state behind on marriage equality
* Lawmakers could vote on gay marriage over the next week
* Bill needs support from one more senator to pass (Adds detail on vote, bill)
By Dan Wiessner
ALBANY, June 14 (Reuters) - New York could become the sixth U.S. state to allow same-sex marriage next week after Governor Andrew Cuomo introduced a bill on Tuesday that needs support from just one more senator to pass.
Cuomo, a Democrat in his first year in office, had vowed to make same-sex marriage a priority during the final weeks of the legislative session. Lawmakers break on Monday for a lengthy recess.
The state-by-state battle over gay marriage has become one of the most contentious U.S. social issues ahead of the 2012 presidential and congressional elections.
"For too long, same-sex couples have been denied the freedom to marry, as well as hundreds of rights that other New Yorkers take for granted," Cuomo said in a statement.
Connecticut, Iowa, Massachusetts, New Hampshire, Vermont and the District of Columbia allow same-sex marriage and 10 states allow civil unions. The first legal same-sex marriages in the United States took place in Massachusetts in 2004.
A recent Siena poll found 58 percent of New Yorkers support same-sex marriage.
New York's Democrat-dominated lower house Assembly has easily passed a bill legalizing same-sex marriage several times in recent years. But the move was rejected the first time it was voted on by the then Democrat-led Senate in December 2009.
The state Senate is now controlled by Republicans and currently 31 of the 62 senators have publicly indicated support.
Senator Ruben Diaz, a Pentecostal minister, is the only Democrat of the party's 30 senators who does not support same-sex marriage, while Senator Jim Alesi was the first Republican to announce his support for Cuomo's bill.
Alesi said his 2009 vote against gay marriage was "political" and "anguishing."
"If you live in America and you expect equality and freedom for yourself, you have to extend it to others," he said after meeting Cuomo on Monday.
Republican Senator Roy McDonald said on Tuesday he would also support legalizing same-sex marriage, telling local media he was "trying to do the right thing."
If the bill passes, same-sex couples could start marrying in New York 30 days later. The bill does not compel any member of the clergy to conduct same-sex marriages.
"From the fight for women's suffrage to the struggle for civil rights, New Yorkers have been on the right side of history. But on the issue of marriage equality, our state has fallen behind," Cuomo said. (Writing by Michelle Nichols; Editing by Eric Walsh)
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