The judge always has the power in a civil case to order that any proceeding commence, or proceed, without the other party present. This, in it's simplest form, is what is at work in a "default prove up" (when a defendant has never appeared, and a default is taken, and the plaintiff then given the opportunity to prove damages). Now, whether or not that would stand on appeal would depend on just why the judge made that order. If it was in accordance with the Code of Civil Procedure in the jurisdiction, and supported by precedent based on the facts of the particular case, then that decision is likely to be upheld, even though the prejudice to the other party is evident. The jurisdiction and whether the diver's waiver would apply to these facts are wholly different questions. Apparently this is a wrongful death case brought against the husband by the wife's family. Jurisdiction is not dependent on where the death occurred, but the husband's domicile. "Wrongful Death" is not something that could be waived in a normal "diver's waiver," since it is not based on the diver's own actions, but the actions of another, whether negligent or reckless, although reasonable legal minds could differ on this question based on the issue of causation.
