Global hyperbolicity is certainly not enough.

Consider the causal diamond
$$ D = \{(t,x) \in \mathbb{R}^{1+1} | |t|+|x| < 1\} $$
in 1+1 dimensional Minkowski space, which is certainly globally hyperbolic. I claim that this set does not admit the so-called "global proper-time foliation".

Observe that with a proper-time foliation, the total elapsed proper-time must be at most 2, this being the length of the longest time-like geodesic you can fit in the causal diamond $D$.

Supposing such a foliation exists with the global time function denoted $\tau$ (so that the level sets of $\tau$ are Cauchy, and $|\mathrm{d}\tau|_g = 1$), and is at least $C^1$ smooth, we can start with an arbitrary point $p\in D$ and follow the flow of $p$ by $\mathrm{d}\tau^\sharp$. By the first step we know that the total length of the flow-line is finite, and by definition this length is non-zero. Call it $\epsilon_p$.

Claim: there exists a point $q\in D$ such that *any* time-like curve $\gamma$ through $q$ must have the total length of $\gamma\cap D$ be strictly less than $\epsilon_p$.

Proof: Consider the point $q = (t = 0, x = 1 - \delta)$ for some $\delta$ to be specified later. Since time-like geodesics *maximize* length among time-like curves, the maximum length that can be attained by a curve $\gamma$ contained in $D$ through $q$ is bounded above by the lengths of the two segments joining $q$ to $(t = 1, x = 0)$ and $(t = -1,x = 0)$. By taking $\delta$ arbitrarily small we see that this maximum length can be made arbitrarily close to zero.

Steps 2 and 3 above contradict, since by assumption the flow of $q$ by $\mathrm{d}\tau^\sharp$ should have the same length as the flow of $p$.

The key is step 3: the same argument carries through whenever your global hyperbolic spacetime $D$ satisfies the condition that "for every $\epsilon$ there exists a point $q\in D$ such that no timelike curves through $q$ in $D$ has length greater than $\epsilon$."