Language itself is culture, not science. Grammar is a science because there are established rules and constructions; however, it is not an exact science as there are too many variables, and the rules even change at times.
I agree that there are a lot of reasons to study language. A would also say that there are many different approaches to be taken i the study of language. As mentioned above there is more than one definition of language. I think for educators it is important to understand the development of language skills in children and to understand the difficulty that can occur when these language skills are not acquired.
In addition to all the great points that have been made, I'd like to bring up how language is closely linked to culture. Different languages, dialects, and even accents have links to traditions, cultural identity, and even power relationships. These are things that cultural anthropologists, sociologists, and psychologists all track as they study the ways that human beings interact with one another.
Also, depending on your definition of "language," the term may also encompass methods of animal communication. Granted, animals don't create sign-based language forms like our own systems of words or characters, but there is evidence of sophisticated communication between members of animal groups in species like dolphins, whales, elephants, chimpanzees, and even bees! Here's where biology and linguistics intersect--100% science, and 100% fascinating!
We can study language and syntax and see that there are prescriptive "rules" about sentence structure, parts of speech, spelling, subject-verb-pronoun agreement that are pure fact. Science tends to have a factual or rule basis to it, and a study of some components of language shares that same quality.
Language is a communication tool that is developed in humans through the intervention of both nature and nurture. This being said, the part of language acquisition that is occurs through nature consists on a number of specific processes that are age-related and dependant on the developmental acumen of the individual.
It is also a highly-cognitive process that can be measured quantitatively through psychometric testing, and analyzed qualitatively through psychology. Therefore, the social component of language acquisition certainly aids the process, but it is the biological nature of the activity that takes place as it occurs calls for the inclusion of different fields of science to explain how exactly it happens.
Language is a science because there are prescriptive (formal) elements, as well as informal elements that must be considered. Like science, the study of language is based on facts, data, and statistics. Regions also affect the use and development of language and language systems. Language is objective not subjective, and all of this supports studying language as a science.
To some extent it is a science because there are scientific ways to describe the parts of language. There are, for example scientific terms for the various sounds that can be made by human vocal structures. The ways in which the sounds used in words change over time can also be studied scientifically -- we can see that languages tend, for example, to drop certain sounds off the endings of words over time. In ways like this, various components of language and the change in those components can be studied in objective ways. This makes the study of linguistics a science.