Phrased in the most general and interesting way (to me):
Given an Earth-like planet with all the right ingredients (e.g. large bodies of liquid water, iron-nickel partially-liquid core, etc.) with one exception—no history of carbon-based life forms—would we find significant amounts of limestone like we have on Earth today?
There are too terrestrial limestones formed on lakes and rivers called tufa.
"Recently it has been demonstrated that microbially induced precipitation may be more important than physico-chemical precipitation. Pedley et al. (2009) showed with flume experiments that precipitation does not occur unless a biofilm is present, despite supersaturation."
source: wikipedia (see quoted paper)
There is no evidence of such deposits on Mars neither. Carbonates are found , but being magnesite (MgCo3) the main carbonate with little percentages of calcite.
So, I can be wrong, but my answer to "significant amounts" is no, knowing little deposits of CaCO3 could thermodinamicaly been formed on a non-life rocky planet.
If we venture out of the "sedimentary limestones", there are also carbonatites, which are igneous rocks crystallised out of a carbonate rich magma. These are rare on Earth (we know of only one volcano that erupts this stuff), but could be more common on other planets. Weathering of these rocks can produce proper sedimentary limestones.