The sedimentary register shows on Earth firsts significant amounts of limestones came from stromatolites and foraminifera at the end of Precambrian.
After life explosion, two main carbonated deposits are registered:
- Powerful basin deposits at the coast with coral reefs.
- Big benthic deposits with foraminifera and coccolithophoridae at seabed
A limestone uses to be those skeletal fragments, that are cemented by CaCo3 at diagenesis, but not always. At calm environments, mudstone can be formed, primarily from algae. Another exception to this are crystalline limestones that have changed their texture at diagenesis and skelets are not recognizable, but that doesn't mean origin of CaCo3 wasn't organic.
Dunham classification of carbonates. Image from: commons.wikimedia.org
There is another exception, as noticed: oolites. Oolites are inorganical concretions around a bioclast or clast. What I have read is this structures are formed at the beaches thanks too to algaes.
Anyhow, a non-life rocky planet can produce carbonates at his first magmatic phases. The rock is called carbonatite, wich is a magmatic rock, not a sedimentary limestone. You could then have some scattered sedimentary deposits where cement is CaCo3. I find it unlikely, but erosion of carbonatites could lead to true limestones.
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
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.