Wasn't a war over the status of Aquitaine still coming, regardless of whether the English king had a claim on the French throne?
A war? Certainly.
The war? Certainly not.
The HYW as it was only developed as it did because Edward III (and later, his great grandson also) was able to make a strong claim to the throne of France.
However, as long as the King of England holds land of the French crown, there is a better than average chance of war. The French king cannot have a vassal beyond reproach, just because said vassal holds other lands outside of France. At the same time, the English king cannot subject himself to the will of the King of France, just because he possesses land in the French kingdom.
At the risk of sounding like a medieval version of
Yes, Minister, it's a question of hats. As far as the French crown is interested, Edward's hat is a ducal coronet. While he wears that hat, he holds lands of the French king, is a vassal of such, and must act accordingly. As far as Edward is concerned, his hat is the crown of England. While he wears that hat, he is a king, whatever the domain, and he is subject to no other king.
This can be seen in the case of Robert III, Count of Artois: one of the proximate causes of the war. He fled Philip's realm and laws, and threw himself on the justice of the King of England. Philip demanded that the Duke of Aquitaine (his vassal) expel the rebellious Count of Artois (another vassal), or - if possible - hand him over to the French crown. Edward, as king of England, can't have the French king telling him who can and can't be allowed at his court, and the refused. Philip, as king of France, has every right to tell his dukes who they can have at their court. The fact that the Duke is also a king is irrelevant. This is an internal French matter. This was one of the primary reasons Philip cited for attempting to confiscate the duchy.
Now, this scenario didn't have to lead to war. A less confident Philip might not have pushed the issue. A less confident Edward may have given Robert some money, and sent him off to Scotland, or somewhere else. But, even if this particular incident is resolved without war, that doesn't solve the long term reason for both the war and the argument over Robert.
As long as the English king wears both hats, there will be wars over the conflict of interests between the two hats. There may be periods of peace, but war will only be one argument over jurisdiction away. The only way for that to end is for the King of England to lose his ducal coronet(s), or gain the French crown.