These are some notes to complement previous answers.
Your concerns are sound: a rigorous definite way to rule out CTCs has not been found. What we have is arguments (and quite nice looking ones) to illustrate that every known universe with CTCs looks unphysical.
Second, there are two nicely-written pedagogical letters written by Kip Thorne addressing your question ,. They mainly focus on physical aspects of the known CTC solutions, and three popular mechanisms that could prevent CTCs: violation of the averaged null energy conditions (the first argument cited in the post), classical instabilities of chronology horizons, and quantum field instabilities (following the notation of , section 4). Although he does not seem to believe in CTCs personally, at the end of  he states that this is still an open question:
It may turn out that on macroscopic lenghscales chronology is not always protected, and even if chronology is protected macroscopically, quantum gravity may well give finite amplitudes for microscopic spacetime histories with CTCs .
 Friedman J 1992 in Proceedings of the 4th Canadian Conf. on General Relativity and Relativistic Astrophysics eds G Kunstatter et al (Singapore: Word Scientific) pp. 183-199.
Finally, regarding the argument against CTCs that uses logical paradoxes, which has already appeared in the post: it is not clear to many people whether CTCs inevitably lead to causal paradoxes. Several studies have pointed out that causal-paradoxes of time travel could disappear once one takes quantum mechanical effects; or maybe their meaning could simply change ,,,. For instance, in the framework used in the first reference the grandfather's paradox does not violate causality. In connection with this, although it is known that some of these models of CTCs , lead to counter-intuitive collapes of computational complexity classes, this is not exactly the same as a causal paradox.
This post imported from StackExchange Physics at 2014-06-11 15:06 (UCT), posted by SE-user Juan Bermejo Vega