I'm not sure I see a "battle" going on, but there is a reluctance on Ginger's part to just comply with whatever Fred suggests (shrugs and eye rolls when he states it's a lovely day). She makes sure he knows that her agreement is an individual thing and things won't progress unless he takes note of that. For its time, this is a refreshingly "modern" take on male-female roles. Compared to the earlier musicals (primarily the Keeler-Powell flicks) the leading lady is no fragile ingenue, she's a woman with opinions, suspicions, perhaps a past -- and yet she's still a desirable and intelligent woman