If a 911 operator is allowed to tell people they don't have an emergency, why can't the emergency room?

The ER is obligated to check you out if you sign in as a patient because of possible litigation in the future.
There is a possibility that although you may be in there for a non-emergency type of problem that there is an underlying cause to your ailment which is, in fact, an emergency.

And of course there is the EMTALA which states The Emergency Medical Treatment and Active Labor Act (EMTALA)[1] is an act of the United States Congress, passed in 1986 as part of the Consolidated Omnibus Budget Reconciliation Act (COBRA). It requires hospitals to provide emergency health care treatment to anyone needing it regardless of citizenship, legal status, or ability to pay. There are no reimbursement provisions.
EMTALA applies to “participating hospitals.” The statute defines “participating hospitals” as those that accept payment from the Department of Health and Human Services, Centers for Medicare and Medicaid Services (CMS) under the Medicare program.[2] “Because there are very few hospitals that do not accept Medicare, the law applies to nearly all hospitals.”

The 911 operator has a list of what is and is not an emergency.

Because of potential litigation. The number of times I’ve been woken up at 3am to x-ray a kid who has been coughing for 2 weeks defies belief. But we do it, because if we discharge a properly sick patient, the name of the hospital will be blackened all through the media for weeks to come, and there will be a legal settlement to be made. In an ideal world, we would be able to tell them to take a concrete pill, but that’s unfortunately far from reality.

So why isn’t the police department held to the same standard? Why can someone on a telephone line decide for you what constitutes an emergency, but a medically trained professional sitting in front of you can not?
I get what you are saying, because a drunk lady came to the ER because she ‘didn’t feel good’ but couldn’t tell us more than that. Turns out she was very, very sick. If that same drunk lady slurred that over the phone to a 911 operator they would get irritated and hang up on her.

@GloPro I think it is something to do with the level of effort on the part of the person involved. It is very easy to make a phone call and talk rubbish, while it takes more effort for most people to make it to a hospital. Getting in the car is harder than making a phone call. Therefore we must assume the situation is potentially more serious, because the person has done more to call attention to their perceived problem.

Nah, that’s not it. An ambulance is sent to most 911 response scenes. Easy transport if the phone call is taken seriously.

Is an ambulance sent to scenes where the caller is thought to be talking rubbish?

That would be the choice of the 911 dispatcher, and now we’re back to the premise of the OP.

I think my point stands. There is every possibility that a 911 (or equivalent in other countries) caller could be making a prank call. But almost no one would make a prank hospital visit, even though they may visit for trivial reasons. It isn’t a matter of the scale of the response, but the probability of misdirection on the part of the caller.