The chaos in Syria has its roots in a severe 4 year drought that displaced upwards of a million people. That inflamed already tense ethnic rivalry and challenged the incompetent and corrupt government that ha existed for decades. Drought has been a factor that brought down civilizations throuhout history, including the Maya and the Chaco Canyon people in North America.
Any kind of disaster has a possibility to threaten global stability. If there was a huge drought then not only would we have major problems with clean water supply but major food shortages would cause global hunger. There would be a complete breakdown of society and possible rioting as people try to take care of themselves and make it to where they can survive.
The issue of water heading through the 21st century will continue to grow in importance every day. There are finite water supply resources, and things like drought can have severe consequences. Local farmers lose crops, ecosystems lose vitality, and even the area of land impacted will be significantly changed when shorelines and such get displaced.
Global stability is threatened by drought. Drought is always going to be an issue but the planet has a unique way of repairing itself without the need for human interaction. Drought causes a lot of problems worldwide and will always be a factor for us to have to consider to deal with.
I think that we are going to find future world wars will happen over water resources, more than they will for oil or political reasons. So yes, when there is a major drought, that will become an event that will have a major impact on global stability. Hopefully, we can learn to work together and avoid those problems.
No, global stability is not threatened by drought, because droughts come and go as a matter of weather. Drought is a weather event. Some years will be bad, some will not. Droughts usually do not affect so much of the globe that they can cause stability. Droughts are usually only regional problems.