In short, the selection of federal judges is done by the executive branch of government. For example, the President is responsible for appointing Supreme Court judges. The idea of a republic, in the pure sense, is to have a representative government... a government that reflects the will of the people. That said, some would say that having the executive branch simply appoint a federal judge would violate the principles of republicanism, as the people have no direct say in who is selected. However, (and this speaks to your question about a democratic process) this is in fact indicative of republicanism. The people do not have a directly democratic response to the executive selecting the federal judge; however, they have an indirect democratic response... that is to replace the executive branch (to say nothing of the legislature that must approve all judicial appointments) during the next election cycle. That is textbook republicanism.