Gemini DBAccess does not include the JDBC drivers --- the only exception is Derby, for which the driver is included.
That means, you need to add the H2 JDBC driver to the target platform.
If you do not want to create a P2 update site out of the JDBC driver, deselect the option "Include required software" when adding Gemini DBAccess. This will result in the property includeMode="slicer" in your target definition file instead of includeMode="planner". Press "Set as Target Platform" again (which also leads to a refresh of the dependency resolution).