Although you can argue this both ways, I would say that the laws were not needed.
The laws were passed mainly because Americans feared that the immigrants would not assimilate and become Americanized. However, it is quite clear that those Eastern Europeans and Italians did assimilate. We no longer worry today about Italians being unable to fit in as Americans. This implies that the laws were not needed.
You could, however, argue that the laws were necessary. You can say the immigrants only assimilated because the laws forced them to. If more and more immigrants had been allowed to come, they could have kept to themselves and never assimilated.
There is no way to prove which of these is right, but I would argue that the laws were not necessary because the immigrants ended up assimilating.