I also experienced a situation like this.
Apparently everything was "ok" with my Tomcat installation, and I also seemed to place the
.war file properly (as I could see its folder being created on the
What helped me find the answer was to check the Tomcat logs; you can see them in the
/tomcat/logs directory (specifically the
My problem was that the version of Geoserver I was using (2.13.0) requires Java 8 to run, and I was using Java 7. I suggest you check your logs, and also the requirements of your Geoserver version. Changing my configuration to use Java 8 solved the problem and I was able to start and access my webservice successfully.