No for the same reasons as England would not turn into a French speaking country*. Actualy nearby France, which will still be rather influential would make it even less likely that Normandy switches languages. I used belgium as an example earlier and can do the same here. Even though Wallonia (only Flandres was French) was part of the Holy Roman Empire, it still retained its Romance language, for a large part because of the influence of France. So the best case situation for an English Normandy would be an Anglophone elite and a Francophone population until nationalism arises. Although, I think in this case france's cultural influence is powerful enough for the elite of Normandy to remain Francophone.
*BTW I want to make clear I was not talking about an England whose king still owns Normandy, I was talking about the Angevin Empire or something similar, in which the king of England controls a large part of France. In that case the centre of power would be in the more wealthy French provinces (for example Richard Lionheart barely visited England and prefered France) . If he would only control Normandy the centre of power would be England.
England was the part of the Angevin Empire that was easiest to tax, and able to provide the most tax - due to the organisation of the country which started with the Anfglo Saxons and continued through Domesday etc. It wasn't because Acquitaine was richer that England that Richard spent more time there - he was born and brought up in Acquitaine, his mother with whom he was very close loved Acquitaine, he was Acquitainian, not English. His father, Henry 2, for example, spent considerably more time in England, to a degree because he had to pacify th country after his accession.
To be fair, I agree that a continued Angevin empire would lead to monarchs spending most of their time in their French territories - but because these would be the most threatened and restless territories (with France after the Vexin, Acquitainian nobles naturally restless etc).