There is a scenario wherein some questionable activity has take place by a member of an organization. The administrator of the organization's Google Apps account was able to retrieve this user's emails and information by simply logging in since he maintained the usernames and passwords.
What states permit such activity and allow such obtained information to be used in a court of law as evidence?
All of them; it could be freely used by the organisation as evidence in court.
The information on an organisation's Google Apps account belongs to the organisation: not the user.
It's legal if this is a work related Google Apps account, and if the admin has been asked by the organisation to investigate.
It is most likely that the admin would get into trouble if he or she just reads stuff on your Google Apps account out of curiosity (and that is covered by the title of your question). For example, your Google Apps account might contain confidential customer data that you received from a customer under NDA. Your organisation could decide to read this data and call their legal department to check if this is a breach of the NDA or not. The admin doing that out of curiosity would be in trouble with your company, but in that situation might also be stealing someone's trade secrets, which would be criminal.
For criminal evidence, anyone holding evidence can be asked to hand it over to the police. If a company wants to avoid this, the only way is to not hold that information.
