The deontological argument is basically the opposite of the utilitarian argument. It states that the needs of a person are highly significant and that it is unethical to harm a person - even for the good of others. One cited example is that of a gunman who has the person being questioned plus ten other people held as hostages. The gunman has told the person being questioned about ethics that he will kill all ten of the other people in the room, unless the person being questioned kills one person. From a utilitarian standpoint, killing one person would be better than allowing all ten of them to be killed by the gunman. However, from a deontological standpoint that would not be the case. Instead, it would still be wrong to kill one person, because that is an unethical act. It would not matter what the gunman did, because that unethical act (killing ten people) would be on him and the person being questioned would not have any reason to have guilt over the outcome.
Naturally, that kind of scenario seems much more severe than whether a family should have access to a deceased family member's email. Still, the same kind of rules apply. Should a service provider who made a contract with a person do what is allegedly unethical and break that contract simply because the person is deceased? The contract was still valid, and the agreement was that the emails of the deceased person would be deleted and would not be kept or transferred elsewhere after the person's death. That included the idea that the emails would not be able to be accessed by others. It is understandable that the family is grieving, and that it would likely help them feel closer to the deceased family member and provide them with some closure, but business and personal are not the same and should be kept apart. The Internet service provider is a business, and had a business relationship with the deceased person. That contract, along with all the other contracts the provider still has with living people, must be honored so that a level of trust is maintained.
The lesson here is really not one of ethics, but one of practicality. If people are concerned that this will happen to their loved ones and they want their loved ones to have access to their email when they are gone, leave email and password information in a safety deposit box, a safe, a file cabinet, or other place, and leave instructions in a will or other document as to the whereabouts of these things. That way, people who may want and need closure later will be able to have that without fighting in court for something they feel they are entitled to but which the courts deem otherwise. Ethics, like morals and opinions, are subjects on which not everyone agrees. That is very common, and something to be addressed in families so it can be kept out of the court system.