It would be even better if IntelliJ IDEA respectively the debugger does not focus on the own browser only. Many people run Internet Explorer to use web applications. IE is pretty different in behaviour than Firefox e.g. and the developers must cope with that situation. It should be possible to debug an application from inside IntelliJ IDEA on Internet Explorer, too.
Some thoughts about a possible solution: The user/developer can start the web application e.g. via context menu "start with" > (a) Internet Explorer (b) FireFox (c) xyz respectively with the standard browser on the individual desktop - and the debugger works (while using one of those standard browsers from outside the IDEA distribution). Some kind of plugin architecture could be useful.