While the identification of Ur in the book of Genesis with the Sumerian city of Ur was popularized by Leonard Woolley, the archaeologist who excavated of the latter, the question remains unresolved. Indeed, most traditions place Biblical Ur somewhere in northern Mesopotamia, outside of Sumer proper (though certainly within reach of Sumerian traders and colonists). But let's assume for the sake of argument that Biblical Ur was indeed the famed Sumerian city of Ur.

By tradition, Abraham was the ancestor of the Israelites and the Ishmaelites. Abraham's nephew Lot was the father of the Moabites and Ammonites. Other relatives are described as Arameans. These are all Semitic peoples (in particular, West Semitic). Abram (the original name of Abraham) is a Semitic name.

"Sumerian" was the name given to the non-Semitic inhabitants of Mesopotamia by the Semitic Akkadians. The Sumerian language, which was completely unrelated to the Semitic languages or to any other known language, was the first language to be recorded in Mesopotamia and remained a classical language for a long time, but ceased to be spoken by around 2000 BC. Insofar as we can place Abraham at some point in history, it is likely that by his time Sumerian was no longer a spoken language in Ur or anywhere in Mesopotamia. "Ethnic Sumerian" might have already become an anachronism by the time of Abraham.

Abraham's clan were nomadic pastoralists. Abraham certainly moved around a lot in his lifetime, staying in Egypt at one point. We shouldn't draw firm conclusions about his ethnicity based on where he happened to be born.

Edited for grammar.