In general, no. Let’s assume that the increase in inequality is driven by new technologies, such as automation, or by foreign trade. Imagine that Chinese competition lowers American middle class wages but gives Apple another export market and thus simultaneously boosts the returns to capital. For our analytical purposes, the new foreign trade is a “new technology” of some kind or another, so doing trade or technology as the cause of the higher inequality should not make a big difference.
Assume also, as many models do, that capital is more mobile than labor.
In many settings it is then the mobility of capital that determines the domestic wage, not immigration. If you keep out more immigrants, that just means capital leaves your country for India or China. Alternatively, letting in more low-wage immigrants limits outsourcing (or automation, as you wish) and keeps more capital in the United States. It may even boost the number of jobs for native-born Americans, who perhaps drive trucks to and from the factories where the immigrants work. Here is some evidence on that point, hardly conclusive but certainly not running against immigration.
It is instructive to look at the polar case. Let’s say American wages were completely determined in global markets. Letting in more immigrants wouldn’t affect those wages at all.
Immigrants also keep their beneficial economic effects in increasing returns to scale models, with or without high inequality in the domestic wage structure.
There are many different ways you can slice this cake, and I am not suggesting the mechanisms outlined above are always the dominant ones. Still, they should disabuse you of leveling the immediate knee-jerk charge that higher domestic inequality weakens the economic case for additional low-skilled immigration.
There are two further points of import. First, if permitted immigration is so high that labor is more mobile than capital, the argument for limiting low-skilled immigration to help domestic workers may become stronger. Second, the “political and cultural externalities” arguments against low-skilled immigration are still on the table.