Part of the issue with answering this question is that "Semitic" is a language group, which was then applied to the (sometimes, but not always) related ethnicities that speak them.

But in this case, that caveat isn't important, since the answer is no. Jews are related to other Middle Eastern groups. This includes non-Semitic-speaking groups, such as Armenians and Kurds, whose ancestries are quite similar to Jewish ones (which illustrates the issue of defining ethnicity by language--you're forced to tweak it with evidence, and then the terms no longer match up. But I'm not sure anyone would call Armenians or Kurds Semitic). Jews' ancestors were among the ancient ethnic groups whose ancestral language was a Semitic one. So were many Arabs (an ethnic group which subsumed many other Semitic ones) and other Middle Eastern groups. The closest thing to a true statement you can get out of that is that Jews are the only speakers of a Canaanite language, but that's a subset of Semitic languages (and Semitic languages also include Arabic, Aramaic, Modern South Arabian languages, Amharic, etc). But in terms of a (retroactively defined) Semitic ethnicity? It's definitely not just the Jews who are the descendants of that.

Sources:

• Haber, Marc et al. “Genome-Wide Diversity in the Levant Reveals Recent Structuring by Culture.” Ed. Scott M. Williams.* PLoS Genetics* 9.2 (2013): e1003316.
• Hammer, M. F. et al. “Jewish and Middle Eastern Non-Jewish Populations Share a Common Pool of Y-Chromosome Biallelic Haplotypes.” Proceedings of the National Academy of Sciences 97.12 (2000): 6769–6774.
• Nebel, Almut et al. “The Y Chromosome Pool of Jews as Part of the Genetic Landscape of the Middle East.” The American Journal of Human Genetics 69.5 (2001): 1095–1112.