I can refute your argument by challenging you to prove that the property 'irrefutable' exists. What does it mean to be irrefutable, and how can you prove it exists? Consider the following.
Your argument seems rather close to Cartesian certainty that begins with "I think, therefore I am". In the same way one cannot think about it and deny thought exists, one cannot use language and deny language exists, at least at first glance. But let me suggest the following.
You claim language exists. So be it. What is it and why is it objectively real? For instance, it is well known that the sounds to convey language do not entail the experience of language any more than the wavelengths of a stream of photons entail the experience of sight. Red is not a property of an apple, it is a construction of a mind, and likewise, words are not properties of sound, but are also constructions of the mind. When speech recognition in natural language processing began, the first problem was determining how sound could be segmented into words because words are not physical properties of sound at all.
Now, your argument, like Descartes's shift to the question, does the mind exist? Unlike an apple, which a strong argument exists from empirical lines of thinking, that it has physical properties, that others can agree in a blind study on those properties, that the apple can be measured by instrumentation independent of human observation, the existence of thoughts and language becomes more difficult.
What is a mind? Can anyone witness another mind? Is there physical evidence of minds, and how do the constituents of minds, whatever they might be, relate to the physical world? How does one even define mind begin with? Here, the questions about "mental phenomena" become far less certain. Cognitive science, of course, has amassed an empirical framework to deal with these challenges which I would argue is rather persuasive to declare a mind exists, and that minds and languages are related. But it begs the question why we should be allowed to use the verb 'to exist' with things we cannot directly measure, see, experience with our senses directly; language is no more physical than redness.
This dispute over the language rules that governs the use of the phrase 'language exists' therefore is a complicated matter because it requires an adequate definition of language and existence, and neither of those tasks are straightforward. My personal belief is that "irrefutable" is tough to apply to any argument, because it implies there is some undeniable truth that is not subject to revision or question, and it's hard to see how that can apply to any claim. Consider the simple argument that we are actually nothing more than simulations of people in a quantum computer in an actual physical universe. If we are a simulation, then not only does it seem language is not real, but it begs the question is our simulated physical universe even real? That's enough doubt to make your claim refutable.
For more explanation on how the property 'irrefutable' itself might not exist, see the IEP's article on fallibilism.