Put simply, no, the devil (or Satan) does not act independently – he is not outside of God’s sovereign control. There are three points to make here which will make this clearer.
Firstly, God is in complete control of everything, and has absolute power over Satan. We see this most clearly in the first two chapters of the book of Job. Satan is unable to do anything to Job without God’s permission, and nor can he do anything to us without God’s permission.