Carla Katz heads Local 1034, Communication Workers of America, which represents state employees.
State GOP Chairman Tom Wilson said the suit was filed under the Open Public Records Act.
"Jon Corzine should stop acting like he's above the law," Wilson said. "The people have a right to know whether or not his personal relationship with Ms. Katz unduly or inappropriately influenced Jon Corzine's actions as it relates to the recently approved public employee contracts."
Corzine's personal relationship with Katz, which ended before he ran for governor, has previously caused him problems. During the campaign, he acknowledged lending her thousands of dollars to buy a house and then forgiving the loan.
Corzine's Ethics Advisory Panel concluded last month that the e-mails did not need to be made public.
Assemblyman Joseph Cryan, the state Democratic chair, told the Star-Ledger of Newark the Republican suit is "nothing more than an act of political voyeurism wrapped up in a meaningless lawsuit."