I set up a second virtual host, but did not configure a Rails context for it. I set up the same External App, Script Handler, and Java Web context configuration as the other virtual host. For this second host, the index.html file was returned when accessing the "/examples/" URI. So I think the problem in the first virtual host was a conflict between the Rails context and the Java Web App context.
This second virtual host setup still fails to handle the JSP and servlet examples in Tomcat. Still getting a 403 response when I try the HelloWorld servlet example. Note that this servlet runs fine when I access Tomcat directly with "http://localhost:8080/".