In order to answer this question, we need to look at what Turner said the frontier did for American society. In a nutshell, he argued that it made American society more of an egalitarian melting pot and that it made Americans more democratic, more aggressive, and more innovative. If these characterstics came about because of the existence of the frontier, then it stands to reason that the closing of the frontier would cause them to fade gradually away.
By that reasoning, the closing of the frontier would make the US an industrial nation like those of Europe. It would be a nation in which people were not particularly individualistic or entrepreneurial. It would be a nation where the various white ethnic groups no longer had a means of assimilating and becoming one people. Finally, it might even threaten American democracy because it would make for a more hierarchical and divided society.
The degree to which this has happened or is happening is, of course, a matter that can be debated.