The claim is ridiculous in one sense and correct in another. It's ridiculous in one obvious sense ... there are written records in types of Arabic that precede the Qur'an by centuries. There are also a number of post-Mohammed Arabic texts and inscriptions, which interestingly enough never mention the Qur'an or Mohammed until you get to around the year 690.
So in no way was Arabic created by the Qur'an.
If you change the wording, however, to say that *Classical Arabic* is, in large part, an attempt to make sense out of the Qur'an in the context of the linguistic/theological ideals and tafsir of the 9th/10th centuries, thereby contrasting Classical Arabic with the varieties of spoken Arabic and Arabic-ish vernacular/dialect prevalent throughout the Middle East at that time, then I'd agree. The real question is how much of a connection Classical Arabic has with the language(s) that the Qur'an's basic rasm was originally written in.
On that subject, I agree entirely with Guillaume Dye's position in this fantastical article:
http://www.academia.edu/4730102/Traces_of_Bilingualism_Multilingualism_in_Quranic_ArabicSpecifically on the subject of Classical v. Qur'anic Arabic:
"The Arabic of the Qur’ān is certainly not identical with Classical Arabic (which I take more as a socio-linguistic label than as a strictly historical one), and some aspects of its grammar which strike us as a bit strange may simply reveal linguistic usage, not always congruent with the later standardization of Classical Arabic grammar (even if Qur’ānic Arabic, as we know it, is partly the result of the standardization of the language represented by the rasm), in some part of the Arabic-speaking world, at a particular time."
Further,
"We should also remember that Qur’ānic Arabic may not necessarily be as homogeneous as generally assumed – and this should be no surprise. Qur’ānic Arabic, of course, is the Arabic of the Qur’ān – a tautology, which should not hide, however, two significant points. First, there is a probable hiatus between the language represented by the rasm (closer, at least in part, to the vernacular), and the language represented by the qirā’āt, which display the influence of the poetic language. Moreover, the Qur’ān, strictly speaking, is not a book, but a corpus, namely, the gathering of relatively independent texts, which belong to various literary genres and are, in several ways, somewhat heterogeneous (for example, the style and vocabulary – see the numerous hapax legomena – of the many “oracular suras” at the end of the Qur’ānic corpus are quite different from those of the other parts of the Qur’ān; more generally, the literary and stylistic quality is uneven)."
Topic: The Quran in English