I understand how your feel for your daughter, but I have to say that even the 'explaining' should be done by YOU, not the lifeguard.
Educating children is the responsibility of the parents, not anyone else. To be honest, when this thing happened, you should have immediately explained to your daughter that why the lifeguard was so serious about it, and why it is your fault not the lifeguard's. Instead, you nurtured your daughter's feelings instead of using the opportunity to educate her with what's right what's wrong.
It's still not too late for YOU TO DO THE EXPLAINING. Tell her that the lifeguard is only trying to help guard her safety and it is YOUR fault for not helping her comply with the pool rules. What do you expect the lifeguard to explain? saying 'I'm sorry for being loud, I was trying to help'?? This should be said by YOU.
Maybe it's hard to hear but I totally agree with what Single Cat said.