SINCERELY,
If the asking and refusal of password could cause breakup between Lovers, then they have only being playing themselves for all the time they spent together.
Well if I was the girl, I would ask for the reason he needs my password & that would determine my next action.
If I am satisfied, I would give him on the condition that he woukd give me his.
But If I was the guy, I wouldn't have ask for password rather I would use the green grass route to check out whatever I want to know.
To me none is at fault base on the fact that before your lover would refuse you anything, It depends on manner of approach.