If you are trying to find some calcite in Minecraft, you will need to first locate an amethyst geode. These are large rock layers that most notably hold amethyst shards, an essential resource for making spy glasses and tinted glass blocks.
That said, you can find amethyst geodes spawning after Y=70 underground and around bedrock levels. You can also find the geodes spawning occasionally on the overworld, exposed both in beaches and underwater.
Geodes will also have an outer layer of smooth basalt. Past the smooth basalt is where you can find calcite just before you get to the amethyst layer. If you want to mine calcite, you will need to have at least a wooden pick or higher.
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Once you got the calcite, that will be all you need to do, but what can you even do with calcite in Minecraft? Well, oddly enough not much, as it seems there is no practical use beyond just using it as a block in your various building creations in the game.
So we honestly don't recommend going to too much trouble getting it unless you are also after the shards you can find in geodes.