One cannot be pro-socialism and pro-immigration. Immigrants are usually needed for private sector jobs, usually either highly specialized or low paying ones. Despite the prospect of a low paying, tough job, immigrants come to America and other non-socialist countries because they believe their children will be able to have a better life. But most union and socialist jobs are reserved for citizens, not immigrants. Thus, the sina qua non of the immigrant story is a large private sector rather than a large government sector. In other words, someone who is pro-immigrant must be capitalist, not socialist, assuming that socialism means a large government sector.
If you're still not convinced, please listen to Milton Friedman:
[Ch 7] "one of the paradoxes of experience is that, in spite of...historical evidence, it is precisely the minority groups that have frequently furnished the most vocal and numerous advocates of fundamental alterations in a capitalist society. They have tended to attribute to capitalism the residual restrictions they experience rather than to recognize that the free market has been the major factor enabling these restrictions to be as small as they are...the purchaser of bread does not know whether it was made from wheat grown by a white man or a [black man], by a Christian or a Jew. In consequence, the producer of wheat is in a position to use resources as effectively as he can, regardless of what the attitudes of the community may be toward his color, the religion, or other characteristics of the people he hires.
Furthermore, and perhaps more important, there is an economic incentive in a free market to separate economic efficiency from other characteristics of the individual. A businessman or an entrepreneur who expresses preferences in his business activities that are not related to productive efficiency is at a disadvantage compared to other individuals who do not. Such an individual is in effect imposing higher costs on himself than are other individuals who do not have such preferences. Hence, in a free market they will tend to drive him out...
[Ch 1] As this example suggests, the groups in our society that have the most at stake in the preservation and strengthening of competitive capitalism are those minority groups which can most easily become the object of the distrust and enmity of the majority--the African-Americans, the Jews, the foreign-born, to mention only the obvious...[Yet] instead of recognizing that the existence of the free market has protected them from the attitudes of their fellow countrymen, they mistakenly attribute the residual discrimination to the market."
Bonus: Our minds tell us, and history confirms, that the great threat to freedom is the concentration of power. Government is necessary to preserve our freedom, it is an instrument through which we can exercise our freedom; yet by concentrating power in political hands, it is also a threat to freedom. Even though the men who wield this power initially be of good will and even though they be not corrupted by the power they exercise, the power will both attract and form men of a different stamp.