No, this is not a golden-ratio spiral. Its closest relative is the Archimedean spiral, whose fundamental equation is $$r=a+b\,\theta.$$
This is the spiral traced out by the water thrown out by a horizontal sprinkler as it rotates: because its horizontal velocity is constant, the radius $r(t)$ of a given drop at time $t$ increases linearly with $t$, whereas the angle it propagates on is the direction of the sprinkler when it was fired, which also increases linearly with $t$; hence, there's a linear relation between $r$ and $\theta$.
Image credit: Anton Croos. I can't find a picture taken from above the sprinkler - apparently people are more careful with their cameras than you'd think.
In the case of your image, there is the additional action of gravity to deflect the raindrops, so the spiral will not be perfect, but the principle is the same. It's important to note that Fibonacci and golden spirals operate on a different principle and they're very hard to sustain over multiple turns, as the radius grows exponentially. This is easy to do with, say, a mollusk that eats more as it grows, but it is hard to accomplish with purely kinematical phenomena.
Kinematical phenomena do, on the other hand, more or less routinely produce archimedean, or archimedean-like spirals. My favourite is this one, which is produced by shock waves propagating at constant speed through a planetary nebula, and produced by the gas emitted by one of the stars in a closely-orbiting binary pair:
That image's now got a very close contestant, and I couldn't resist posting it here. This one is sort of, roughly more or less the same: it was produced by gas venting out of a half-spent rocket stage from a Russian ICBM which was rotating as it moved.
Image source here; explanation in Phil Plait's Bad Astronomy blog here.