Finally, I was able to solve it.
I added a file called server-update.config to the .ebextensions directory placed at the root of your .war file. The contents of the file looks like this:
container_commands:
replace-config:
command: cp .ebextensions/server.xml /etc/tomcat8/server.xml
Further information about ebextensions: https://aws.amazon.com/blogs/aws/customize-elastic-beanstalk-using-configuration-files/
Regarding the server.xml, I take the server.xml placed at /etc/tomcat8 and I added the following section inside the <Host> section.
<Context path="app" docBase="ROOT"/>
Thanks to this answer: https://stackoverflow.com/a/18226607/3576460