My cat's eye ruptured. Is surgery to remove it the only solution?

Hello. I am sorry to see that John is having issues with his eye. It does appear that the eye is severely compromised, and if the white clear discharge seen is the material from inside his eye then yes, it needs to be removed. You can always see an ophthalmologist for their opinion, however if that discharge is from a ruptured globe there is likely very little they will do aside from remove the eye. If the discharge can be 'wiped away' and a normal eye is underneath (unlikely based on what your vet has said and the picture provided) then it could be saved, however this is unlikely. Good luck with John, and remember, animals with 1 eye do exceptionally well, so don't worry about John only having 1 eye, he will do great!