If there is a very rude/angry customer in a store, what can I (as another customer) do to have this person removed from the facility?

You do nothing. You stay out of it. This does not involve you except as a looker-oner.

I worked for social security and we dealt with rude and angry people constantly. We were trained to deal with them. We called security only if we were threatened. We did NOT expect and DID NOT WANT another member of the public to involve themselves. If the store employee or store manager wants the person removed they will follow their procedures.

Just leave the store. It isn't your problem.

nothing. it isn't your area. do leave the store ... in case he loses it, you don't want to be there

The best you can do is find a security guard, if one isn't in the vicinity. If one is, you can suggest that the guard keep a close eye on the situation.

However, unless the customer is being threatening, presenting a danger, there isn't much to be done -- certainly not by you. A salesperson could say, "Ma'am, I don't want to calla security guard but I will if necessary," and a security guard could step in and say, "Ma'am, if you can't calm down and be polite, I will have to escort you from the premises," but that's not for YOU to say.

Alerting a guard is as far as you can go, unless someone brandishes a weapon and makes clear threats, in which case you should leave the store and call the police.

Call the manager.

You can complain to a manager but that's it. You're a customer. With zero power.

Complain to the manager on duty, You cant really do anything other then that unless they assault you. The manager can have them removed from the store.

Nothing.
The facility is private property and you have no standing.
You can CALL security, but this is like me being in your home and upset about the actions of another guest at your party.

As a customer, there's nothing you can do but call the police and make a complaint about the other customer.

That's not your place and the staff should call the police. You can stand between the bad guy and the store clerk or simply stand very near allowing the implication that you would lend assistance to the clerk if the situation appears likely to get physical (but do not throw the first punch).

Complain to the staff

Nothing really, other than reporting to management and hoping they kick them out.

you can try calling the police , they might remove them