Mental Health Foundation chief executive Stuart Robinson criticised that view as "simple-minded and unacceptable", and this afternoon Nash responded - taking to Twitter to say "I got this wrong".
"Should have consulted the experts first," he wrote. "My sincere apologies."
The Police policy lists current use of psychotropic medication, such as medication to treat depression or anxiety, as a medical exclusion.
Prime Minister Bill English was steering clear of the issue at his regular post-Cabinet press conference, refusing to say what his view on the policy was.
"That is a matter for the Police...it is a matter for them, I'm not going to try and run the police force from here."