Could it be argued that maybe whites have become more conservative in reaction to the changing demographics? There are several European countries currently with similar-ish demographics to the proposed hypothetical which are more liberal than the US.
You beat me to the same thought. Without the idea of social programs going to "others", you might find whites more willing to be supportive, thinking that they will be the beneficiaries. The great social welfare improvements of the twentieth century in the United States occurred during the New Deal, New Frontier and Great Society, all of which fell during the 20s-60s pause in immigration, as did the great progress in the civil rights movement postwar. You could argue that with civil rights, a white majority felt more secure in its majority such that it was willing to confront a particularly ugly aspect of its society. Once immigration opened up again, a period which coincided with the decline of the industrial economy, the white majority began to feel economically and culturally besieged, which gave rise to a white backlash.
One thing is for sure: a restriction on immigration from 1965 through 1985 would have probably prevented a lot of the economic blame casting that was directed at immigrants and minorities.
The downside, of course, is that US GDP today would almost certainly be lower. The literature is pretty consistently clear that there are huge economic benefits to immigration. This is not to say that those benefits are necessarily fairly distributed, a factor which can feed a backlash. The 2016 election should provide some evidence for that: high immigration + high inequality = resentment of minorities and immigrants and elites.
Sort of off-topic and tangential, but I've always wondered why immigration opponents haven't called for a pause in immigration rather than permanent restriction. A pause of ten to twenty years for assimilation and social adjustment sounds a heck of a lot better and less xenophobic than stirring up resentment of all immigrants.