Bomhard & Kerns (1994) included Sumerian in Nostratic (which would make it a distant relative of both Indo-European and Semitic, among others), but none of the other Nostraticists have included it. Shevoroshkin, Illych-Svitych, and Dybo thought it might fit into a very old, very widespread hypothetical macrofamily called Dene-Caucasian, which would also take in Basque, North Caucasian, Yeniseian, and Na-Dene, and maybe Etruscan. Even though I follow the work of these scholars, I honestly cannot see Sumerian fitting into either of those proposals even with a shoehorn. Joe Greenberg, for all his work on macrofamilies, still considered Sumerian to be an isolate.
Gratuitous ethnic slurs against Arabs are ignorant of the fact that ancient Arabs were known for matriarchal polities, of which the Queen of Sheba is a familiar example. The earliest mention of Arabs in writing, in Babylonian records c. 500 BC, recorded them as matriarchal. Arabs a few generations prior to Islam had a society where women had much greater sexual freedom and legal rights. Islam at first was a force to redress the imbalance of patriarchal power that had recently caused much social inequality, and allowed women to own and inherit property in their own right, which did not begin to occur in Europe until the 19th century (and American women could not get credit cards in their own name as late as the 1960s for comparison). This is not the thread to go into such a complex matter, but just to note that this facile presentism about Arabs is unhelpful. Somewhere in Kim Stanley Robinson’s Mars trilogy is an analysis of ancient history that contrasts the gender equality of Minoan civilization, which died out, with the male supremacy of the warlike Sumerians that persisted in subsequent civilizations, just for a contrasting interpretation of history. Anyway, social norms have nothing to do with language families!
Turkish nationalists of the early 20th century convinced themselves that the Sumerian language was related to Turkish, so they claimed the Sumerian civilization for the Turks. I sure don’t think anyone today still takes that seriously, but there are traces left with various Turkish institutions named after Sumer. Likewise, I’ve noticed Hungarians who claim Sumer for the Magyars with no more basis in reality than the Turks. Who wouldn’t like to claim the world’s earliest civilization? Specious claims abound.
All that said, I have to concur with what everybody said above that Sumerian is an isolate for all that we can tell. Certainly modern southern Iraqis must have a degree of Sumerian genetic ancestry, but in the intervening millennia all kinds of other ethnic groups like Iranians, Kassites, and many kinds of Semites got in there too. The area that is now Kurdistan corresponds pretty closely with ancient Urartu and Mitanni, who spoke Hurrian—another isolate.