H2 is fundamentally an RDBMS (Relational Data Base Management System), so it's absolutely not "object oriented". It happens that its implementation is written with an object oriented language, namely Java, but it has strictly no consequence on its concepts.
That would similar to think that as you used a steel tool to shape a hammer in wood, the hammer is indeed steel...
As mentioned fge in its comment, to manipulate your database, currently managed by H2, in an object fashion, you need a specific library named ORM (for Object-Relational Mapping), for example http://www.hibernate.org/