Not in precisely the way you mean, but that can be done: https://www.sqlite.org/expridx.html
Postgres can index the entire JSON document (or parts of it) and they can support unknown query condition (e.g. using a JSON path). This isn't as fast as a proper B-Tree index, but still faster than a full table scan.
I believe this could probably be done using a trigger on document insert, but that would involve actually inserting each path and value into it's own table, rather than SQLite generating it on the fly, so it would likely more than double the storage requirement and require inserting potentially hundreds of rows for each document, depending on what your original document structure looks like.