No, it's a longitudinal study (it looks at several existing groups with different levels of facebook use and gives them a survey to measure life satisfaction). This can extablish correlation, not causality. The causality can be completely reversed from their hypothesis - unhappy people use Facebook more.

Only an experiment, doing an intervention on a group, can establish causality. You stop eating for two days, you're starving - hey, not eating causes starving, who knew ? In this case, if you can make low-Facebook-using groups to use a lot, and vice-versa, and measure the effect of this intervention, that would establish causality. However, you can imagine the practical difficulty of ensuring people change their habits. They rarely do it when it has a positive impact on their health, they certainly won't do it for an experiment. This is called patient compliance in medical jargon.

They draw a causal link when, technically, they have a correlation. Combined with other studies which they cite, they have a compelling argument, but if causality requires something like a double-blind, they don't have it.