You had not done anything wrong in confronting him. As far as should you continue with the relationship, that is something you have to decide. The factors that will play a role it his are-
do you think you can trust him
are you willing to stick around and see what kind of a guy he is
and will you be ok with yourself if he turns out to be unreliable/untruthful.
It is normal to give someone a chance but after that, they ought to prove themselves.
Musicians do travel a lot and if you are always going to wonder what is taking place while he is away, then that is not healthy.
In addition, it does not seem appropriate that after he had started dating you that he should still keep his profile on that website and encourage other women to keep in touch with him.