The debate regarding whether language or culture is primary in the sequence of human experience has a long history with many an attempt to refine the definition of terms and so to technically approach the matter. In addition to specialized speech on the subject science has likewise weighed in with so-called empirical data. Given the complexity of the question (as well as the limited space and time allotted in the present circumstances), one could do worse than begin on the level of common sense, and so I will largely avoid scholastic or academic jargon and evidence. Most simply put, culture is the broader category, and so the primordial experience for human beings. Allow me to explain.
Inasmuch as human beings do not spring up ready-made from the earth, they are necessarily surrounded by others, and so are part of a family, or a community, and a larger whole. Not unlike other gregarious beasts, humans emerge into a shared world, one that consists of patterns (whether behavioral or otherwise), rules (whether explicit or implicit), habits for survival (whether learned or innate), and so on. All gregarious beings continue to share in and shape this world. The point is that regardless of how much things might change in the shared world of human beings, there is always a degree of continuity in human experience. And among other things, what is continuous is a sense of that which is shared.
Even the most reclusive and silent of human beings (and so of animals) is the product of two others. To put it differently, there has never been a true Robinson Crusoe, for he too had a mother and father and some basic experiences held in common with others; therefore, a community, and an experience of sharing reality at large with others is primordial. All of us, regardless of how much we are conscious of it or not (and so this applies once again to all gregarious beings), have at the bottom of our lived experience a sense of the common, or the "that-which-is-held-between-us," and so the shared. In its most general sense, culture refers to this sharedness, and there is a degree of necessity to it given that we are not strictly products of ourselves. Nature is the ultimate cause of this continuity and connectivity, and so we can say that human beings are naturally cultural beings.
It is interesting to note that the first use of the term "culture," in the latin as "cultura," implied the "tilling of the land," and its root, "colere," referred to "tending, guarding, cultivating, and tilling." No doubt the original meaning has its origins in an agricultural experience, and so an engagement with the larger world and the work that allows humans to nourish and maintain themselves. It seems to me that no matter how culture is defined, and so no matter how much the definition may be refined, one cannot escape the element of things held in common, or what I have called sharedness-and so how groups of peoples respond to the broader world. Using the term loosely therefore means that it may be spoken of in degrees. It may then be meaningfully applied to certain beasts (the precursors of human beings), pre-linguistic peoples, and to more complex human groups. Perhaps a useful separation may then be made between culture and civilization, whereby the latter serves as a term of distinction to denote greater complexity. But alas, that is another matter.
As for language, its most basic element is that of expression. In other words, it is a symbolic representation of some kind, whether of an imperative, instructional, aesthetic, religious, or other nature. Language too, no doubt, is something that is shared, for there are no private languages (at least not in any meaningful sense); however, it is a product of the shared, and not its cause. Nor is it as directly synomous with the shared as is the cultural. The shared, or the cultural as herein discussed, exists whether it is linguistically represented or not. Communities of beings remain communities whether they are linguistically bounded or not. Language certainly has the ability to bring peoples closer together, for example, it makes rules and expected behavior explicit by ways of laws. However, the common, the often predictable (or the patterned), and the continuous are not dependent upon language, especially when discussing things like basic mating or hunting practices, which all would agree are cultural (and all would likewise agree are found among non-linguistic beings as well).
Language can serve to heighten the cultural and even refine it (or civilize it, as based on the above suggestion), and indeed by way of language we may think about whether language precedes culture or not. But before being able to linguistically think such thoughts a being unreflectively finds itself in some kind of common, shared, and so cultural world.