Currently, the web server is not started in the h2.start.sh script. Ideally, this should be incorporated there.
java -cp h2*.jar org.h2.tools.Server -web
seems to do the trick
See http://www.h2database.com/javadoc/org/h2/tools/Server.html#main_String...
Issue fixed and committed on dev branch.
Issue is kept open until it is ported to trunk.
Currently, the web server is not started in the h2.start.sh script.
Ideally, this should be incorporated there.