Very unlikely. I am not any sort of expert on trains, but the general rule in the United States is that people are free to engage in whatever (legal) business transactions they wish. A business can’t discriminate against someone because of their membership in a protected class (sex, race, and so on) - but businesses can refuse to engage in transactions because they don’t think they’d be sufficiently profitable, and they do that all the time. You generally can’t force a business to take your money.
I know nothing about the relevant American laws, but I would note that this kind of principle is often voided when a company has a de facto monopoly and that its services are required for other companies to operate. And trains and railroads are frequently an example of that. It’s quite common that the company owning the tracks has an obligation to “rent” them to other companies operating trains.
I’m not saying it’s the case in the USA, I wouldn’t know. But I don’t think you can assume that the “general rule” applies in this case, or in similar situations like power lines, airports, etc…
Private rail cars were common in the heyday of rail & robber barons. They still exist today in very limited numbers, owned by rich hobbyists.
U.S. railroads today own track & engines. Different companies own most of the (freight) cars. The car owners pay the railroad to haul their cars from here to there. The railroad is in the business of renting its hauling services to anyone with a car needing to move. This goes on every day.
The only issue with a prvate car or cars today is that the railroads (7 nationals, and ~40 regionals) are used to dealing with a small (several doxen) rail car companies each of whom is pretty big. For a railroad to deal with you, they’d need to be set up to sell haulage retail, not wholesale. Administratively that might easily be more trouble than it’s worth to them.
OTOH, to the degree rail is still regulated as a Common Carrier, they would be required by law to accept your request for haulage. Common Carrier regulations are a specialized area of US law & trump the general point raised by Mr. Excellent above. Without digging through the CFRs, my bet is that Common Carrier regulations are in force which would require them to take you.
Odds are, you won’t want to pay for your own crew & engine, so “taking you” means attaching your car(s) to a freight train departing at a time of their convenience. Freight rail is the very opposite of the “time-definite delivery” pioneered by FedEx. More like “We’ll pick your cars up later this week, we thingk, and they ought to get to your destination about 7 days later, plus 5 or minus 1.”
You can do it, but not cheaply or easily. Railcars for people have to meet lots of safety requirements. Doors, windows, brakes, electrical systems all have to be up to snuff. If you make the grade, you can pay Amtrak large wads of money to haul you on one of their routes. (Or insane amounts of money to go where you want.)
If your car does not make the grade, then you can have your car hauled wherever you like, but you can’t be in it when it moves. Also you cannot be in any car on a freight train. (You car is checked out for passengers, but the rest of the train is not.)
When you get there you have to find a place to park. Few places offer such services.
I once fantasied about buying a motor home and a caboose and stuffing one inside the other. Cabooses are insanely cheap. But then you run into the inspection requirements.
Let’s just assume though, that I really am very rich with my own engine and crew. Do railroad companies allow other railroad companies’ engines on their rails, or must one company hook its cars to the other company’s engine once it’s on their rails?
Amtrak can run its own engines across multiple companies’ rails, can’t it? The military used to own & run its own engines, although I don’t think they used the national rail system very often.
I’m sure Canadian, US & Mexican engines must be able to cross borders, right?
In 1989 I worked for a start up company who was going to run a private service for very rich people from Chicago, Philly, NYC, Boston and DC. Then expand it.
They never could make the costs work. Basically it had private or semi-private cars and a lot of luxury.
So the idea is do-able to a degree. The company had to negotiate a lot the only real trouble was the rule “freight rules.” Which meant if there was a bottle neck the passenger train would halt and wait.
Your train for instance, if it got stuck for whatever reason, you could wind up in a mess. The rates you have to charge were outrageous. It was around 10K to go from Chicago to DC (private car).
The idea only lasted around 6 months and never got past trials. There just aren’t enough people willing to pay that (and that was in 1989) and it takes so long compared to a plane. Imagine paying $10,000 and having to wait 2 hours because a freight train is taking priority.
I would bet if money was no object the OP goal is do-able. But it would be astronomical.
I am a fan of old time Radio and people like Jack Benny and “Burns and Allen,” even did shows where they chartered their own train (Union Pacific) and did shows and promos from it. (That was the late 30s / early 40s till WWII put and end to unnecessary travel)