No. Economic growth does not and cannot eliminate scarcity. Nothing that we know of in this world can possibly eliminate scarcity. Scarcity is a condition of all human life, no matter how rich. Scarcity is the basic fact that underlies economics. Scarcity is eternal.
Scarcity comes about because of two factors. First, human wants are unlimited. We people always want more. Second, resources are limited. There are not enough resources in the world for everyone to have everything they want. This is true for everyone. Not even the richest man in the world can have everything he wants.
In the past 50 years or so, America has experienced a great deal of economic growth. We have become immensely richer than we once were. But scarcity still exists. If scarcity did not exist, we could have all the gasoline we wanted, all the iPads we wanted, and all of the tickets to professional sports events or big concerts that we wanted. This has not happened. Clearly, economic growth has not ended scarcity.
But could it end scarcity? No, it could not. Simply think about tickets to the Super Bowl. There is no way that we could ever create a situation in which all the people who want to go to the Super Bowl could do so. Alternatively, there is no way that everyone who wanted to go to Hawaii for vacation in the winter could do so. These resources and many others are scarce and will always be scarce. There will never be anything (so far as we can possibly imagine) that will end scarcity.