If so, how would I do so? And why would patent law allow it?
While I'd say Jay's answer is snarky as well as correct, you were probably looking for a detailed explanation about what you could patent, how you'd go about it, and why it would be allowed. If you were looking for a way to show the system is absurd, prepare to be disappointed.
What you can patent
Yes, you can patent a robust communication protocol. No, you cannot patent all "robust communication protocols". The former is a specific implementation of a protocol, the latter is what many would call an "abstract idea."
And no, you cannot patent the Morse Code because it's been around for a very long time (and I believe, was also covered by a patent in its time.) This is "prior art."
Also, as mentioned in a comment above, funny that you chose the Morse Code as your example, because Samuel Morse did try to claim the broad, abstract idea of "transmitting information via electromagnetic signals", and that was rejected by the Supreme Court.
However, assuming you had a novel robust communication protocol (for whatever you define "robust" to be) or a novel method of implementing it you, could patent those and aspects thereof. You could even patent a new way of implementing an existing protocol if it was sufficiently novel and non-obvious.
For instance, you could patent a method of implementing Morse Code using carrier pigeons instead of electric signals. The 172-year old protocol isn't what's being patented, but your new implementation of it. Anyone else could continue using Morse Code via plain old electromagnetic signals and not infringe your patent.
If your novel carrier pigeon-based method consists simply of attaching chits of paper containing morse code, it's obvious because it combines well known methods of using paper to contain information and using carrier pigeons to transmit information. However, if your method involves, say, training pigeons to peck out the dots and dashes with their beaks on arrival, that's likely non-obvious.
How you can patent it
You would file a patent application describing the protocol and/or method in detail, and including a series of claims identifying the specific parts of the invention you think are novel and non-obvious. Then you (or ideally, your attorney or patent agent) and the USPTO would have a series of back-and-forths where you try to carve out the exact bits and pieces of invention that are novel and non-obvious in light of relevant prior art the USPTO examiner finds.
You file the application and claims, the examiner does a search for prior art and returns with reasons to reject your claims, you adjust your claims to sidestep prior art, and repeat until you end up with a claim that both parties are satisfied with. (You want claims that are broad as possible while covering the essential invention, but the examiner will only allow claims that are not covered by or made obvious by prior art.) Note one of the reasons for rejection is lack of enablement: if you do not describe in enough detail how you get your pigeons to peck out the intended Morse Code, you'd get a 112 "lack of enablement" rejection.
The resulting claims are what will finally determine the exact scope of what your patent covers.
Why patenting something like this is allowed
Because one of the goals of the patent system is to encourage disclosure of inventions while allowing the inventors to profit from the fruits of their (intellectual) labor. As a competing carrier pigeon service, I would be very interested in knowing how you got your pigeons to memorize Morse Code! Think of the benefits! No more paper costs! No more cutting out tiny chits of paper and encoding them with dots and dashes! No more tying little chits of paper to tiny pigeon legs! No more retransmits if the chit of paper falls off!
You on the other hand, don't want me to simply apply your invention to my pigeons (or heck, even just kidnap a few of your pigeons!) and enjoy the benefits of all your work figuring out how to get the pigeons to peck out the right dots and dashes. Alternatively, you may not mind me using your method... as long as I pay you for it. That's where licensing comes into play. Patents can be used as a legal instrument to enforce licensing of technologies and methods. If I don't pay up, you can use your patent to sue me (and that's about all you can legally do with it.)
On the other hand, the system does not want people claiming ownership of abstract ideas as that would truly hinder progress. Which is why what the patent covers will be very specifically defined by the claims. Sending Morse Code using plain old electric signals is not covered. Sending Morse Code by encoding dots and dashes in the flight patterns of carrier pigeons is not covered. Only sending Morse code by having pigeons peck out dots and dashes would be covered.
By filing a patent application with the patent office.
To promote the progress of the useful arts.