"The nurse said that the doctor would come."
Translation:O enfermeiro disse que o médico viria.
chegaria -> would arrive
I guess it could be used too, yes, although normally you'd probably indicate the time frame as well e.g. "O enfermeiro disse que o médico chegaria em breve / em 5 minutos" (The nurse said that the doctor would arrive soon / in 5 minutes").
Without the time frame it sounds a bit weird, even if acceptable. Using "viria" is better when you do not want to commit to a specific time.