Applying the comparative method to contemporary dialects (not MSA) would not result in Classical Arabic, since the contemporary dialects have lost features found in Classical Arabic, such as case. However, parallel to proto-Romance, a proto-language antecedent to the modern dialects could in principle be reconstructed. Ferguson (1959) "The Arabic koine" discusses how the modern dialects probably descended from an intermediate development that co-existed with Classical Arabic. It is generally recognized that Arabic was not a single monolithic language and that there was considerable dialect variation in pre-Islamic times. Classical Arabic was one of those dialects.
If in fact all modern dialects derive from a single spoken Koine, we might be able to reconstruct that Koine – Ferguson discusses some of the common features that he believed characterized that Koine. However, technically he does not reconstruct anything (lay out comparative tables and find the common feature), instead he compares what is presumed to be the older state as represented in Classical Arabic, and comments on how dialects differ. For example he notes that geminate verbs change from the madadtu ~ madda pattern to maddeet ~ madda in colloquial Arabic, not indicating which daughter languages have this form.
There is an article Owens (JAOS, 2003) "Arabic Dialect History and Historical Linguistic Mythology" which discusses some of the problems of dialect reconstruction, and he notes that the paucity of reconstructive studies is a surprise, but the lacuna may be explained by the assumption that modern dialects derive directly from Classical Arabic, so why would you look at a reconstruction based on modern dialects? (The point being that CA and Proto-Colloquial might be sister languages, so you should do reconstruction)
There is a thesis by William Cowan, A Reconstruction of Proto-Colloquial Arabic, Cornell 1960, which is available through Proquest, and which covers phoneme reconstruction in the main dialect areas.