Very few serious economists pin growing wealth inequality on immigration and only immigration. It's not just happening in the US but elsewhere as well. There are a variety of factors in play, and I think a lot of good economists will even admit that we do not know all the answers. Here are some of the other major ones:

* Technology, which will happen with or without immigration.

* Globalization, which will happen with or without immigration. Make it hard for people to immigrate, and companies will eventually open branches elsewhere.

And there may be specific problems with the Borjas 2015 paper in any case: http://www.nber.org/papers/w21801#fromrss