It depends on the monitoring service. Most free monitoring services monitor only if an ip pings, if a proper response code is returned(200, etc), or if http is accepting connection. These services probably won't catch the error.
Some monitoring services do something called string based monitoring. In string based monitoring, the monitor pulls the monitor page and checks it for a string. If the string is present, it either alarms or passes depending on the test.
For example the search string is copyright and you pass if it is found. If the page loads the footer that contains copyright, it passes. If there is a server error and the error message doesn't contain copyright or it white screens, it fails and alarms.
A different example is the search string failure and you want to alarm it is present. If you have a test page, you have it run a couple of test( can i connect to mysql, write a file, is there enough diskspace, etc) then print success. If not print failure and it will alarm. This is good for running multiple tests and having it alarm if only one fails but things like static pages works. I've seen people setup scripts to check server load and fail if is to high or disk space too low to alert someone of a possible future problem before it becomes a issue.
You probably want to try a monitoring service like this.