This is actually pretty hard to do.
I would say the users canonical default browser is stored at HKCU\SOFTWARE\Clients\StartMenuInternet which then leads you to the HKLM key where the real info is. See
https://msdn.microsoft.com/en-us/lib...(v=vs.85).aspx for more details.
The problem is that this is only supported by XP, 2003 and Vista and only Vista provides the full UI experience (IIRC), this means you will need a fallback strategy.
Now we need to know what the definition of the default browser is when the user has not specifically chosen one. Is it the app that handles http? https? .html?
Either way, once you have decided you would then use the Assoc API or possibly the applicationregistration stuff in Win\Com.nsh to find the executable and then start it.
Why do you think you need to open the users browser without navigating anywhere?