Please note that unemployment rate and available jobs are two different things. I have focused on the second, the number of jobs available to US citizens. And there are not a finite number of jobs, but a variable amount depending on business cycle, GDP growth or contraction and other factors.
Nevertheless, we can still examine the relative size of these markets and very recently at that. It would be kind of curious if the rules of supply and demand worked for everything but employment. I would point you to this study that found
Government data show that since 2000 all of the net gain in the number of working-age (16 to 65) people holding a job has gone to immigrants (legal and illegal). This is remarkable given that native-born Americans accounted for two-thirds of the growth in the total working-age population. Though there has been some recovery from the Great Recession, there were still fewer working-age natives holding a job in the first quarter of 2014 than in 2000, while the number of immigrants with a job was 5.7 million above the 2000 level.
All of the net increase in employment went to immigrants in the last 14 years partly because, even before the Great Recession, immigrants were gaining a disproportionate share of jobs relative to their share of population growth. In addition, natives' losses were somewhat greater during the recession and immigrants have recovered more quickly from it. With 58 million working-age natives not working, the Schumer-Rubio bill (S.744) and similar House measures that would substantially increase the number of foreign workers allowed in the country seem out of touch with the realities of the U.S. labor market.
So to answer your question, does immigration diminish the number of jobs available to US citizens, the answer is an unambiguous yes. The number of jobs held by natives is diminishing while the number of jobs for immigrants is rising. While there are not a finite number of jobs, there are a finite jobs at a given time. Therefore immigrants are taking jobs at the expense of natives.