Turing machine + time dilation = solve the halting problem?
Note that Turing's proof is one of mathematics, not of physics. Within the model of a Turing machine Turing defined, undecidability of the halting problem has been proven and is a mathematical fact. Hence, Turing's proof will not 'fail' in the spacetimes, it will simply not prove anything about the relation of the halting problem and time dilation. 

However, what you'll likely want to know is whether a 'time dilation Turing machine' can solve the halting problem.

If you want to study this the influence of 'time dilation' on a Turing machine, you'll have to specify a formal model by which we can formally understand what it means for a Turing machine to make use of time dilation. Unfortunately, this format is ill-suited for providing such a formal model (unless someone else has written a paper about it) as creating the model is far too broad.

However, it isn't unlikely that some formalisation indeed is able to solve the halting problem. This paper by Scott Aaronson, Mohammad Bavarian and Giulio Gueltrini looks at computational models under the assumption that so-called Closed time-like loops exist and conclude that the halting problem is indeed computable within that model.