Race is most definitely a socially constructed thing. There is, biologically speaking, no such thing as race. As the links below show, the idea of race has no real meaning in genetic terms.
Instead, race is a socially constructed thing. We can see this in how we determine who, in the United States, is an African American and who is not. Take, for example, the picture of the three basketball players in this link. Of the three players closest to the camera, two are considered to be African American while the third is considered to be white. This is hard to understand in physical terms because all three of them have skin tone that is very close to being the same. In addition, the two African Americans (#30, Stephen Curry and #11, Klay Thompson) are considered to be the same race as the two players in blue warmups behind them and as the head coach, Mark Jackson, in the suit and tie next to Thompson. Clearly, we base our ideas of race on something other than genes and physical looks.
In the US, we have typically constructed race in such a way that only people who are 100% white are seen as “white” while those who are 25% or less of some other race (African American, Asian) are seen as members of that race. There is no biological reason to conceive of race in this way. It is really just a way that our society has chosen to define race.
It is also worth noting that races have changed over the years. 100 years ago, people who are now seen as white, such as Italians and Greeks, would not have been seen as white by many Americans.
Race has no real meaning in biological terms. It is something that we have constructed on the basis of our societal attitudes.
I'm less interested in what social scientists say about race and biology than what biologists, anthropologists, and other physical scientists say about it. How on earth can physical characteristics like skin color, eye shape, etc., be socially-derived?
This is not a discussion of racism; no one here is arguing for superiority of one race over another. It is a discussion of whether race is related to the geographical origins of one's ancesters. Social scientists, and even many physical scientists, are so afraid of offending that they go through tortuous paths to both validate both sides of the argument. A classic example -- and pretty much a "rubber-meets-the-road" one -- is the article linked below regarding forensic anthropology, a field that does not allow for political intervention into the scientific process.
The following quote from a University of Wisconsin essay titled "An Anthropological Perspective of Ethnicity and Race" should not be dismissed out of hand:
"The concept of race from a physical anthropologist's point of view refers solely to biological variation. This includes phenotypical differences in stature, skin color, hair color, facial shape and other inherited characteristics which tend to include genotype variations."
Obviously, there is no one universally accepted definition of race. To deny a biological connection, however, is to render the discussion meaningless. No race is superior to another, and the equal treatment of all races should be universally accepted practice. I always find it troubling, however, when attempts at avoiding any comment that can be construed as insensitive can be allowed to undermine the intellectual integrity of the debate.
I remember several years ago when an elderly African American educator and friend of mine told me that race is a social construction rather than a biological trait. I was caught off guard. Here is one of my favorite quotes on this topic: "When you begin to understand the biology of human variation, you have to ask if race is a good way to describe that" (Janis Hutchinson, Biological Anthropologist).
I visited a powerful exhibit about race, which is a project of the American Anthropological Association. I am attaching a link to its website, which has information that can help guide your inquiry into this question.
Social scientists do not believe that race is a biological fact. They believe that race is socially constructed. That is why you can have lecture notes like the one in this link, which starts out
Race is constructed. Boundaries are not natural but created.
It is why we can have textbooks like this one, with chapter summaries that say
The use of the concept of race for sociologists is as a social construct; a race is a group of people who see themselves—and are seen by others—as having hereditary traits that set them apart.
Social scientists are essentially unanimous on this point. Race is not a biological fact.
Race is also a product of biological and environmental factors. It is not solely a socially constructed trait.
Race is a physical characteristic derived through genetics. It is entirely biological. Racial distinctions originate primarily from the different geographical regions from which human beings evolved over thousands of years, and tend to be unique to the originating location. Archaelogical studies have determined that the earliest modern man originated in Africa, and was dark-skinned. As man migrated over land masses and evolved or adapted to new environments, he began to take on physical characterists unique to those environments. Consequently, the physical characteristics of people whose ancestry originates in Scandinavia are very different from those of people whose ancestry originates in the ring of countries encircling the Mediterranean Sea, or from those originating in Sub-Saharan Africa.
An important indicator of the biological uniqueness of people from different races is often found in the cellular flaws unique to certain categories of people. Only people who can trace their lineage to Africa, for example, are known to develop sickle-cell disease or sickle-cell anemia. While the debate over whether Judaism is a religion or a race will likely continue forever, it is a medical fact that only Jews -- specifically those from the Eastern European Ashkenazi Jewish population -- develop Tay Sachs disease. While additional "races" have been found to carry certain variations of Tay Sachs disease, the genetic mutation found in Ashkenazi Jews is unique from the other strains.
It should be noted, in closing, that "ethnicity" and "race" are frequently used interchangeably, but are considered two distinct categories of human beings by scientists.