I think it could be done, but it'll take some -serious- planning, mapping and more than one horse. Even if in the old days people were riding across the continent, you have to realize that they weren't abiding by horse-restricting laws, speed limits and asphalt roads dictating where they went. They could go pretty much wherever they felt their horse could take them and they didn't face the hardships of today's prices. They'd ride into a town, be able to find a livery, blacksmith, and an inn, hang out for a few days and leave again for the next town. In today's world, you'd be reduced to finding a local who'd be willing to short-term board your horse and checking into a Holiday Inn.
So I'd think the first logical thing to do to even -see- if this would be possible, via laws, regulations, human crap, would be to sit down with a map of the states and find routes that you would be able to take. Don't necessarily look for just road-side routes, look for maybe old trails that you can take or fields (for example, you can ride through the Pawnee Natl Grasslands from Colorado to Wyoming, never touching a road) that you can use to connect, etc. Really get into the nitty-gritty and do your research. Make a list of the states you want to ride through, look up their laws and regulations regarding people riding their horses on streets, search for boarding barns that you could contact for a short-term board of a few days (that'd require a phone conversation), whether or not the city near that boarding barn has amenities for the average tourist, etc. Be sure you also include vets, farriers, and a human doctor in your research so that you can tend to both your horse and yourself. Think of the worst possible situation and prepare for it. If you can find a horse-friendly route to take with all the amenities that you need that take you through the desired path - it could very well be possible, depending on your funding.
Then you'd have to start the other big part of this. Getting yourself a bomb-proof horse or two, a support system, and your physical condition. These guys are right, you're going to need someone who can follow you, or at least be on call and in your general area, if the worst happens. In the old days if you fell off and broke your leg, you either figured out a way to (barely) survive, or you died. In today's world, we're surrounded with technology that could save your life but wouldn't do you much good if you're in a black-zone. (where cell reception doesn't exist.) Figure out a system of communication, like say.. you have to call and check in with the trailer hauler at least once a day. If you don't call, they should worry and know what general area you'd be in so they can start searching for you. Etc.
Your body needs to be shaped up, toned up and prepared for long rides. Believe me when I tell you that your body would Not appreciate a sudden change from 0 to 8hrs in the saddle daily. You could start with lessons or buying a horse and starting riding in increasingly longer increments. This goes both ways for you and your horse. Your horse should be in the best possibly physical condition and, just like you, would need to become accustomed to long periods undersaddle with someone on his back. It'd take some time to find a really good packing horse like that, in my opinion. You'd need at least two of them (arranging to meet up and trade horses out at one of your destinations) so as to prevent too much stress from one horse. He'd need to be traffic proof, bomb proof, spook free, all that stuff, and a good packer. All of them would be.
You'd need to be proficient in first aid, human and equine, have some basic survival skills in case you ended up camping (shelter building, campfire skills, hunting small mammals) and you'd have to be relatively good at identifying wildlife, mammal, reptilian and of course, insects, so as to prevent getting bitten by something poisonous. You'd have to be able to understand, learn and accept the hunting season and laws of each state you were in, possibly needing a permit in each state or whatever, and you'd have to pack plenty of food (yours and the horses') and water for you both while keeping a good supply of simple camping gear, such as a sleeping bag, matches and a multi-tool (the rest can be found out in the wilderness.) You'd have to be comfortable with this notion of camping, as I'm certain there's not always going to be a road for you to ride on. You'd need to invest in good, high quality tack and clothing for you and your horse. Your tack would have to be specially fit and designed to work best with your horse and prevent at least some stress on his back, and you'd need good thermal clothes in case it were cold, and good cool clothes in case it were hot.
You need to be proficient at reading weather and knowing when a storm is brewing, whether it's a tornado-forming sort of storm or just a bad rainstorm. You need to know what the average climate of each state you're riding through is. (Does it drop to single digits each night? Soar during the day?) You need to strategically plan according to season and climate for each state and what you're comfortable riding in. Your horse would have to be a hardy sort of horse, like a Mustang, instead of something like a Thoroughbred. You HAVE to know your horse's daily limits.
You'd have to study and memorize your planned route, estimating how many days it would take to reach your destination and have a group of people waiting for your arrival. Your horse would be your first priority, getting him checked out by a vet, farrier, and into his boarding facility to rest. (I'd give him about a week before even packing out again). A doctor, a friend and a hotel for yourself, leaving yourself enough opportunity to replenish your supplies before packing out. Taking a week in each destination would be a great way for you to travel around (by Car) and be a tourist too. You need to be fair to your horses and know when they've reached their limit, setting up a way for them to get home without having to be ridden, as well as having your next "duck" set up so you can continue on your trip without too much a delay. You need to have an opt-out plan set in place for yourself if you decide you're done. You also need someone in tune with the current news, who can update you on anything happening with government (closing down natl parks), states (natural disasters) or your family (worst case scenario).
Lots of planning. I, personally, think it's a doable idea. You just need to be smart about it, do your research, and prepare for the worst while hoping for the best. Lol.