Its true that most of the calcite on earth is biogenic in origin, however that’s a selection effect. life on earth , specifically calcareous lifeforms in the ocean, keeps the concentration of calcium and bicarbonate ions in seawater low by using it to produce calcite and aragonite. if the concentration was greater, you’d see a lot more non-biogenic calcite. so while caves and hot-springs absent of calcareous life forms may not be a significant sources of calcite on Earth, its because there aren’t many places on Earth where life doesn’t have a chance to use it all up. And in those few instances were you do have supersaturation of calcium in seawater, you have both biogenic calcite, and nonbiogenic oolites.

As an analogy, lets look at iron oxide. we see lots of the stuff on Mars in the form of hematite and magnetite, and In fact its another indicator that Mars at one point had liquid water, as both form in aqueous environments. however on earth, most of the iron oxide we mine and use comes from “banded iron formations”. BIFs were formed by ancient cyanobacteria, as a means of handling the oxygen they produced through photosynthesis. they took iron ions out of sea water to bond with their waste, leaving behind layer after layer of biogenic rock. we don’t say the the iron oxides on Mars must be biological in origin though, because its clear that in the absences of life, chemical processes could have just as easily rusted surface iron.

I’ve been looking for some since you suggest there isn’t a very high correllation between calcium carbonates and life on this planet, but the only two alternative processes I’ve found are either derivative of biology or are very much enhanced by biological processes:

http://www.astrobio.net/news/modules.php?op=modload&name=News&file=article&sid=2596

http://geology.geoscienceworld.org/cgi/content/abstract/27/7/645