It could........but why would you want to? One of the major appeals of natural stone walls or boulders is that they do look "natural".......or mossy......or even dirty. Although as exposed to elements as those boulders are, they sure aren't dirty :-) More likely to be just natural coloration or some lichen growth.

Is that actually snow??

I don't know about limestone, but I had a rock wall built that used granite boulders in some places to add stability and they get dark and "dirty" looking. I've had them pressure washed (gentle pressure) and they look so much better when clean to me. I love moss covered elements in the garden, but I loved the look of the rock wall when it was first built, so for now, I like it when it's restored closer to that newness.

Your wall looks like it could be used for sitting? If so, clean might be more appealing. Try cleaning it one time and see what you think. If you don't care for the look, it will soon be "dirty" again and you can let it be. Or you could encourage moss to grow and maybe it would replace whatever organisms are just dark and dirty looking.

If you were to see a limestone outcropping in nature, it'd be covered with lichens, moss & mildew and would be quite dark. Here, it's going in that direction (as will be the case) but the fresh cut lighter colored limestone has not caught up with it. Over time, as the light colored limestone catches up and darkens, the overall look will not be blotchy looking as it is now. This is not the kind of feature that you'd want to keep pristine looking. It will be better when it's all dark and natural looking.