Well, As EJA just said, physically such a thing should not be a problem. I think that I could ride 50+ miles per day given the time and inclination without difficulty. It is just a matter of pacing.

The major objections with possible counters are:

1) Do you really want to do such a thing for 20+ days straight? It could get old. But then again, you should know the answer to that one.
2) Communication would have once been a problem but with WiFi so available now, you can stay in touch with folks easily with a decent tablet or smart phone. So that should be no problem.
3) Medical is pretty much the same answer. If some problem does pop up, given your route, you will not be more than a few hours from home assuming you have someone to drive out and get you.

So if it appeals to you, why not? I would suggest that you take a digital camera to help record some of the high points of the trip and keep a journal of sorts. When you get back home, you can combine the photos with the journal for a real cool momento. Have a blast.