Will sharing posts to Facebook, Twitter, or other social media platforms help improve my Google search ranking?

Indirectly, sharing posts to multiple platforms can help drive people to your Google Business listing, which will increase your metrics.

There is no direct ranking benefit to sharing posts with Facebook. It can help if people click from Facebook (or other social media sites) to your GMB post. This increases engagement (clicks, views, reading reviews, etc.) on your listing, a ranking factor.

Twitter also has a special relationship with Google, where Twitter posts get indexed exceptionally quickly. Since Tweets are public, you can also use Twitter to feed data to the Google algorithm for crawling purposes.