"I have personal views that would probably not sit well with my professional occupation right now," he told Ben Standig of The Athletic on Monday. "I think I'll just leave it like that."
Two members of Washington's defense, lineman Caleb Brantley and linebacker Josh Harvey-Clemons, have opted out of the campaign.
Del Rio also discussed his use of Twitter to engage in conversations about polarizing topics, which created controversy in June.
"I am open to discuss at any time with anyone my thoughts and respectfully have exchanges," he told Standig. "It's just not always possible in today's environment. I think it's important to be able to have your own opinion and respect for each other. People may have an opinion that is different than mine, which is OK. I'm OK with actually having a conversation about it with anyone, including my players."
Del Rio joined the Washington staff in January under new head coach Ron Rivera.
The 57-year-old California native spent a decade in the NFL as a linebacker beginning in 1985. He started his coaching career in 1997 and has previously served as head coach of the Jacksonville Jaguars and Oakland Raiders. Del Rio won a Super Bowl with the Baltimore Ravens as their linebackers coach.
Washington is scheduled to open the regular season Sept. 13 against the Philadelphia Eagles.