We typically interface with a RDBMS through SQL. I.e. we create a sql string and send it to the server through JDBC or ODBC or something similar.

Are there any RDBMS that allow direct interfacing with the database engine through some API in Java, C#, C or similar? I would expect an API that allows constructs like this (in some arbitrary pseudo code):

Iterator iter = engine.getIndex("myIndex").getReferencesForValue("23");
for (Reference ref: iter){
    Row row = engine.getTable("mytable").getRow(ref);
}

I guess something like this is hidden somewhere in (and available from) open source databases, but I am looking for something that is officially supported as a public API, so one finds at least a note in the release notes, when it changes.

In order to make this a question that actually has a 'best' answer: I prefer languages in the order given above and I will prefer mature APIs over prototypes and research work, although these are welcome as well.