The protection system will detect the earth fault and trip the generator offline. Generally, this is a hard shutdown (load rejection), not a soft shutdown.
A protection system should attempt to outage only the smallest part of a system as necessary to clear a fault. However if the cost of a protection system can not be justified by the value of the system being protected, than a larger part of the system must be outaged to clear the fault.
Said another way, either the protective settings are wrong, or the generator is to small to justify it's own proper protection.
Or the designer was being cheep, and the customer diden't know any better.
'Load rejection' is just fancy way of describing opening of the generator breaker under load. It's non-specific, for example a non-electrical event such as a turbine load dump could result in load rejection but in that case the generator would normally stay available for a reclosure or re-sync onto the system. Load rejection [≠] electrical trip of generator.
An earth fault on the generator bus would almost always result in load rejection because protection will trip the generator main breaker and the excitation to de-energise the generator bus. Before anyone contends that operation of a resistance-earthed generator with a stator earth fault is possible, I don't know of a operator of big utility-class machines who would be brave enough to actually do it in anything other than the most desperate circumstances. On a small emergency set of a few MW then the benefit of keeping emergency power available outweighs the risk (and potential cost) of wrecking the set in the process.