Depends highly on the actual mandate and how it's realized, and in that regard on how the organization would be incorporated into the international community, including non-members. I'd support a global NATO in the sense of an intergovernmental organization focused solely on the mutual defense of it's member states, yes. I'd say such an organization works much, much better in terms of actual determent of possible attackers, and coordination of it's military assets than a whole bunch of single bilateral treaties. And it's more distinct and less convoluted - WW1 was already mentioned before.
But i doubt such an organization would remain a pure defensive alliance. The actual NATO isn't one anymore, it's used more and more for assignments on a global level and international interventions. I'm not against international interventions per se if they're based on humanitarian reasons, but, as past events have shown, it's often hard to make a judgement in such cases if an intervention is actual reasonable or even more harmful, and of course there is always a risk of "humanitarian" or "democratic" reasons simply beeing a false pretense.
If such an global NATO would be found on the later premise as a kind of "World Police", i wonder how they'd actually word their criteria and regulations for interventional operations. There are even members of the current NATO who've some examples of human rights not-taken-too-seriously in some areas of their own. I imagine it really, really hard to reach an international, binding consens for what allows interfering with another nations' sovereignty, and if a global NATO is specifically intended to take over such tasks, they better have a good foundation for it from the beginning instead of deciding on a case-to-case basis.
An global NATO in that sense would only make sense for me if it is connected to the UNO, and answers to it, instead of beeing a completely seperate organization independent in it's decisions. A seperate military organization of such a scale, not bound to the UN and specifically meant for taking action even in non-memberstates, even for humanitarian reasons, would be a really big step backwards. As a seperate organization with self-given powers as an international police it could actually serve as a kind of rival and competitor to the UN, undermining the latters authority even more, and lead to an even more bloc-shaped world than we have today, because non-memberstates of this new organization would inevitably become alienated and feel threatened by it if they have no means of influence over it's interventional actions via the UN.