Gregoire has shown support for the measure and her signature on the bill would make Washington the seventh state to legalize same-sex marriage in the United States, The Seattle Times reported.
The House debated the bill for about 2 hours before approving it with a 55-43 vote -- Rep. Jay Rodne, R-North Bend, said its passage was a "foregone conclusion." The state Senate passed it with a 28-21 vote last week.
"Marriage is the word our society uses to describe committed lifelong relationships," said Rep. Jamie Pedersen, D-Seattle, a gay lawmaker who led efforts to push the legislation through.
"I would like for our four children ... to grow up understanding that their daddy and papa have made the kind of lifelong commitment to each other," Pedersen said. "Marriage is the word we use in our society to convey that idea."
Rodne, who opposed the measure, said same-sex marriage is not an issue of civil rights.
"Marriage has been the union of a man and a woman," he said. "When that union is consummated there is potential for the creation of life. Marriage is about life."
Gay-marriage opponents have threatened a referendum if the bill is signed into law by Gregoire.