Card says, "your opponent can take one action" and RR p.23 says "Control of the battlefield does not change." Does it mean that opponent can't do claim battlefiled action?

No, it means that once the round ends, the player in control of the battlefield will remain in control of the battlefield, and Retreat itself doesn't change that (essentially, it's the same as if both players pass). There's nothing to stop the opponent claiming, and having control of the battlefield at the end of the round.

Thanks