The claim as it is stated in the headline is not supported by the research. There are studies which show some possible negative impacts on wages for certain groups, but not a general increase in the overall unemployment rate. Here is a review of relevant research internationally (not just in the USA) by Okkerse (2008). He concludes that:
the probability that immigrants increase unemployment is low in the short run and zero in the long run. Most area analyses and time-series analyses fail to find a significant influence of immigration on (un)employment probabilities. See for instance the findings of Ganget al. (1999) and Shanet al.(1999) for the EU and of Simon et al. (1993) and Marr and Siklos (1994) for the USA and Canada. Nevertheless, some studies do find an increase in unemployment rate (Winegarden and Khor, 1991), unemployment frequency (Winkelmann and Zimmermann, 1993) and unemployment duration (Winter-Ebmer and Zweim ̈uller,2000). Both area analysis and time-series analysis produce reasons to believe that if there is an employment effect it will especially hit the unemployed (Winter-Ebmer and Zweim ̈uller, 2000; Gross, 2004). In the long run, immigrants create more jobs than they occupy and unemployment lowers permanently (Gross, 2002).
At least two of the studies mentioned there are about the United States specifically. Among these Simon et all (1993) look at "various cities in various years" and find:
The evidence indicates that there is little or no observed increase in aggregate native unemployment due to immigration, even in the relatively short run during which adjustment frictions should be most severe.
The findings from Winegarden and Khor (1991) are also for the US and include only, "a small, but significant, increases in unemployment among [white] teenagers (of both sexes) and [white] young women". Overall though they emphasize that "results clearly do not support commonly-expressed fears that undocumented immigration has caused any substantial increases in unemployment among the presumably most vulnerable groups in the U.S. work force." Keep in mind the caveat that their data is from 1980.
I have emphasized overall unemployment rates as this is what the headline question asks about, but keep in mind that there may other negative effects like reduction in wages and that these effects might be limited to specific populations, particularly other immigrants.