Republican state Rep. Bob McDermott told Abercrombie, a Democrat, in a letter Friday he would seek an injunction against the law if the bill passed both chambers, the Honolulu Star-Advertiser reported Tuesday.
The state Senate voted 19-4 to approve a marriage-equity bill Tuesday, four days after the state House approved the bill 30-19.
State Circuit Court Judge Karl Sakamoto said he would consider the law's constitutionality.
McDermott said a 1998 state constitutional amendment giving the Legislature power to define marriage as between heterosexual couples supersedes the new same-sex marriage statute. He argued a popular referendum was now needed to redefine marriage.
Sakamoto raised questions Thursday about how voters in 1998 knew the amendment would later authorize the Legislature to approve same-sex marriages, the Star-Advertiser said.
The state Supreme Court said in a landmark 1993 ruling denying same-sex couples marriage licenses violated the state Constitution's equal-protection clause.
The U.S. Supreme Court held in June the federal Defense of Marriage Act, which let states refuse to recognize same-sex marriages granted under other states' laws, was an unconstitutional violation of due process and equal protection. The court left it up to the states to decide whether to legalize gay marriage.
"I believe this bill provides equal rights for all people, is legally sound, and is in accord with the Hawaii state Constitution," Abercrombie said Tuesday, after the Senate vote. "I look forward to signing this significant piece of legislation, which provides marriage equity and fully recognizes and protects religious freedoms."
The governor, who supports gay marriage and called lawmakers to a special session to consider the bill, did not say when he would sign the bill.
But aides and lawmakers, including Democratic state Sen. Clayton Hee, chairman of the Senate Judiciary and Labor Committee, said the signing would be Wednesday.
The law is to take effect Dec. 2, letting lesbian, gay, bisexual and transsexual people, including tourists, get marriage licenses in the state.
Clergy would have a right to refuse to perform gay weddings, under the approved compromise measure. Churches and other religious groups would be allowed not to provide goods, services and facilities for same-sex weddings and celebrations if they considered the events a violation of the groups' religious beliefs.
The Hawaii law's Senate passage came a week after Illinois lawmakers passed a same-sex marriage bill.
Democratic Illinois Gov. Pat Quinn said he would sign that bill Nov. 20.
Another 14 states, the District of Columbia, eight New Mexico counties and seven American Indian tribal jurisdictions, covering a third of the U.S. population, issue marriage licenses for same-sex couples.
The other states are California, Connecticut, Delaware, Iowa, Maine, Maryland, Massachusetts, Minnesota, New Hampshire, New Jersey, New York, Rhode Island, Vermont and Washington.
Notable deaths of 2014 [PHOTOS]