OMG Sorry though I know it says resolved here, I just wanna post this here cause I tried using this forum post to solve this same issue I have and it typically shows here that know one was able to fix the issue resorting to the point of reinstalling your OS makes no sense at all and I figured it out on how to fix it all by myself. Hard to find a solution here in the forums.
How to fix protocol handler for osu when clicking on "osu//edit" links.
Try to go to Registry editor by typing in "regedit" in windows key+r command window.
In HKEY_CLASSES_ROOT section, find the "osu!" key.
Click on that and you should find 2 keys.
One would be for the osu! beatmap association and the other one would be the URL Protocol.
If URL Protocol data section is blank, copy the location of your osu launcher and paste in there by double clicking URL
Protocol.
EX; E:\Computer Files\Programs\Games\Osu!
Other than that if it won't work, try to do a clean reinstall of osu (or course backup your stuff first).
Or
Reinstall your browser.
Or
Clear stuff in the FF: Vista/7: C:\Users\<USERNAME PCmu>\AppData\Local\Google\Chrome\User Data
Or
Create a new windows user for a clean start.
Last resort would be reinstall your OS but I wouldn't do something that stupid just to fix something so minor.
But if you don't wanna go and reinstall your browser for this, you can just drag the osu!link into osu itself and it should
read it without a problem though it's something that kinda hinders on your modding sessions, it works though.
These are optional though coz I already tried these as people said they helped based from what I read in the forums though
they didn't work me until I did the changes in the registry.