The answer below is not correct. Upon further reflection, I believe that the correct causality condition is indeed global hyperbolicity and not the weaker stable causality.
I believe this is a causality condition known as stably causal. You can read about this in Wald's "General Relativity" Chapter 8, in particular his Theorem 8.2.2.
A spacetime $(M,g)$ is stably causal if there exists a continuous nowhere vanishing timelike vector $t$ such that the spacetime $(M,\tilde g)$ with $$\tilde g = g - t^\flat \otimes t^\flat$$, t^\flat,$$ where $t^\flat$ is the dual one-form to $t$ relative to $g$, has no closed timelike curves.
Wald's Theorem 8.2.2 says that this is equivalent to the existence of a global time function on $M$.
Theorem 8.2.2 A spacetime $(M,g)$ is stably causal if and only if there is a differentiable function $f$ on $M$ such that its gradient is a past directed timelike vector field.
There is a whole hierarchy of causality conditions in GR. Global hyperbolicity implies stable causality and this in turn implies strong causality.
Global hyperbolicity might be too strong.