Is weak sustainability (Hartwick's rule) possible on a global scale?

First of all Hartwick's rule has originally been derived under rather specific assumptions, among others a Cobb-Douglas production function and constant population and technology. Therefore copying it one on one to the real world would require a huge leap of faith.

Although it has later been generalized to broader classes of production functions, people have also pointed out that instead of a prescriptive rule (invest resource rents in man-made capital to realize constant consumption), it is in fact a descriptive result. If we observe constant utility then resource rents must have been reinvested.

Asheim, Buchholz and Withagen (2003) show among others that it is perfectly possible in a world with constant technology and population not to have zero net investment, and still constant utility over time, as well as situations where we follow the Hartwick rule and have non-constant consumption (and hence non constant utility).

Blindly following the Hartwick rule, on a global scale or on a regional scale for that matter would therefore not be a good idea. That notwithstanding several papers have shown strong links between savings in the past allowing for constant or even growing consumption in the (far) future (e.g. here, here and here).

Intuitively it makes sense that saving rents from non-renewable resources to make-up for lost production capacity is a good policy measure. Whether or not it will be enough is a different matter, and very much depends on degree of substitutability and technological progress.