I may be wrong here as I'm not a legal expert, but I know of no law requiring the postal service to conduct or deliver a service that requires that deliveries be made on Saturday. If there are no laws that obligate a 6 day per week delivery schedule, then no law has been broken.
Unless there's something I'm completely unaware of (and I try to keep up on my legal stuff), they're not doing anything illegal. The postal service will never be completely eliminated because there are some services they do that electronic services simply cannot, but a lot of what they do is being phased out. This is the answer they have at the moment, while it isn't a popular one there's nothing legally wrong with it.
I saw on the news this morning that the postal service is still going to operate on Saturdays. It will still have its offices open for public use. Most importantly the main focus is to keep the mail running. They are still going to deliver medications and important first class mail, they just are not going to be delivering things that are not important, things that can afford to wait until Monday. They have to make money and cannot wait for the Federal government to take action.
The United States Postal Service is not breaking the law by eliminating Saturday mail delivery. For one thing they will still deliver packages on Saturdays so anything of importance can be shipped in that way. Saturday delivery is also not all that important as most businesses are closed on the weekend. This is a step in the right direction for the Post Office, if they do not cut back they will be forced to close for good.