The move, announced by the league on Wednesday, means Brown will miss the Giants' season opener against the Dallas Cowboys on Sept. 11.
Brown can still participate in preseason games and practices. The league did not announce what Brown had done in violation of the policy.
According to documents obtained by NJ Advance Media, Brown was arrested in May 2015 on a misdemeanor domestic violence charge. He was booked into jail and released on $2,000 bond.
According to the police report, Brown grabbed the victim's wrist during a quarrel while the victim was picking up a phone.
"While I do not agree with the suspension, I will accept it," Brown said in a statement released by the Giants. "I have exhausted the appeals process and have no other options along those lines.
"I will continue to work hard for this team, and I have tremendous confidence in my teammates and in my ability to move on and contribute to the team."
Brown is entering his 14th season in the league, and he has spent the past three seasons with the Giants. He made 30 of 32 field goal attempts last season.
The one-game suspension will cost Brown $72,058 in base salary. He also loses out on $15,625 as part of his per game roster bonus, so the suspension will cost him a total of $87,683.
The Giants have two kickers currently on their roster, with Tom Obarski and Brown. Obarski, a second-year player out of Concordia State who has never attempted a kick, could replace Brown against the Cowboys.
Information from ESPN's Jordan Raanan was used in this report.