Washington won't be eligible to return to the active roster until the day after the Cardinals' fourth regular-season game.
The 26-year-old former second-round draft pick -- named to his first Pro Bowl last season -- admitted he had made "a mistake." Exactly what that mistake was has not been disclosed.
"I take full responsibility and I understand that I let down my teammates, the organization and fans," Washington said. "I apologize for that and promise that no one will work harder to make up for it."
Washington posted a career-high 134 tackles, nine sacks, two forced fumbles and an interception in 16 games for Arizona last season.
"One of the biggest things we preach is accountability; Daryl knows that and our team knows that," Cardinals Coach Bruce Arians added. "We will deal with this and others will have to step up until he's back. I'm confident they will."