Pry also has the meaning of to look closely or inquisitively.
Consider a case where your spouse orders your Xmas gift, to be delivered to him by mail, and then says to you Don't pry into the package when it comes,
The idiom uses a verb before "prying eye": have a prying eye, keep a prying eye on, etc. Calling someone a prying eye invites a collision with "private eye," i.e., a private detective hired to cast a prying eye.