Can a US physician prescribe medications to non-US residents?

As the two other answers mention, no problem except for the absurd prices that you may encounter if you don't have a good medical insurance. Use https://www.goodrx.com/ to minimize drug cost, and if you only consult a physician to get a prescription from them, then you can do it online for a typically much cheaper price (eg 50 USD at https://www.goodrx.com/, but there exist other similar online services).

FYI: Given the name of a US pharmaceutical drug, how can I know its equivalent in other countries?

Other answers have already covered that you absolutely can visit a healthcare professional, be prescribed drugs, and purchase them.

The notes about cost and coverage are pretty important.

Travel Insurance

A travel insurance policy is a very good idea for this reason; be sure to look carefully at the coverage provided like dollar and time coverage limits, but also whether prescriptions are covered at all (some plans do and some don't). Additionally, most of these policies of course have provisions against covering "pre-existing" conditions so if these are regular medications you already know you need, for an existing condition, they probably won't be covered in any case.

Local Coverage

While buying local coverage directly is often ridiculously expensive, there are subsidized options available for temporary residents sometimes.

You mention staying for an "extended period" but not via citizenship or permanent residency, so I assume "extended" for you means either as a tourist or on a non-immigrant visa such as a student (F) visa.

Some states do have medical coverage options for immigrants and non-immigrant visa holders. Options typically require that you are legally present and documented.

I know New York state best so that's the only examples I will give, but it will be helpful to look at the states where you are staying.

New York state

New York State lays out its immigration requirements for public coverage in this document. You would fall under Category 4 most likely, and thus be ineligible for anything but emergency care.

However, New York also has a health care marketplace called New York State of Health, and one of the options there for low/no-income applicants is called the Essential plan, and non-immigrant visa holders (such as foreign students) are eligible for this plan.

This page lays out various coverage options for immigrants (it's a New York City website but I think it applies to New York state generally).

More details about the Essential plan itself are in this document.

While the marketplace generally has an open enrollment period (a time of year when enrollment is open, and otherwise not available), the Essential plan can be enrolled into at any time.

Assuming you meet the income and other eligibility criteria, your monthly cost will either be $0 or $20. Other fees like co-pays (a small amount you pay for each doctor visit, or each prescription drug) may also be included, or may be $0 (anecdotally, those who I've helped apply for the program have paid nothing ever for any of the coverage, however they also had no income at all).

Unlike the health coverage most Americans have, this plan also covers dental and vision.

The one thing to be aware of is that this plan will only cover in-network providers. That means that only healthcare providers who have a pre-existing contract with the insurance company who backs this plan, will be covered by the plan. Depending on where you are from, this system, like much of US healthcare, will be confusing and probably surprising. The important thing to note is to check that any place you are receiving care accepts not only this insurance company, but the specific plan (it will not be accepted as readily as some other plans, even other plans from the same company).

This is not usually difficult to determine when you are seeing a single provider, and planning it in advance; it can mostly be a problem when going to a large hospital, especially for something unplanned like an emergency, because in large hospitals, not everyone you see at the hospital is part of the hospital for billing purposes, and may be out of network. I won't belabor this point more but it's something to research if you go this route.

Yes. You don't need a passport, green card, or even valid visitor status to receive medical treatment in the US.

The business side of US medicine is a huge mess, so you'll want to have good insurance. The ease with which you will be able to find an appointment will most likely depend on where you are and on the kind of doctor you want to see.

If the medication is a controlled substance, I suppose it's possible that there might be requirements relating to identification or something like that, which could be a matter of state law, but your valid foreign passport ought to suffice if such a requirement does exist. (There is such a requirement for the sale of pseudoephedrine, for example, even though a prescription is not required.)

The doctor's location matters. Not the patient's.

A doctor licensed in California can practice medicine on any human physically inside California at the time of care.

Doesn't matter where you're from.

In an emergency/rescue they will turn a blind eye, but normally cross-state doctor licensure is complicated.

If you go to an emergency room, they cannot refuse you care. They are going to give you whatever care is required to make you dischargeable, i.e. medically stabilized and with a post-discharge care plan in place.

Anywhere else you go, there will be a conversation about money before care is rendered.

In the emergency room, the conversation will happen at some point prior to discharge, and care -to-discharge will not be contingent on the answer.

Yes, the system has a vulnerability where people without insurance will ignore a simple infection that needs a $90 GP visit and $3 antibiotics, and present at the ER gone fully septic and needing a $100,000 ICU stay. Uninsured generally can't afford that, so hospital eats it.

As such, many cities, certainly all the large tourist cities, have programs of one kind or another to get the uninsured in front of a GP to get that $3 scrip. It's cheaper. An example is a "free clinic". So if you need GP care, that is an option.

Also, parts of the medical system are geared to work with cash-and-carry customers, and set reasonable prices accordingly. For instance Quest will do lab tests and Planned Parenthood will do STD testing. Lots of people do COVID testing, and it may take a little chasing for a foreigner, but most states will get you vaccines for free through one program or another. Very, very few people in America pay for COVID vaccines.

Some foreigners believe US healthcare is the best in the world. US immigration is very happy to have medical tourism from the foreign rich, as it props up struggling hospitals. In fact, the absurd "sticker prices" of medical procedures are for that demographic (they certainly don't expect domestic uninsured to pay it lol). However Immigration is on the watch for poor foreigners planning to steal healthcare by reporting to an ER and dodging the bill. They would refuse entry to such people.

If you need medical care as a foreigner, you'll get it. And that includes getting medication.

Of course you may or may not have it covered by whatever travel and medical insurance you bring with you, which may make the purchase extremely expensive.

Haven't been to the US recently, but last time we were there my dad had to go to the ER and got treated like anyone else, and got the medication he needed as well like anyone else. Only difference was the way it's paid, as we had to pay for it out of pocket (rather than have it billed directly to the insurance company) and later declare the fees to our insurance once we got home.

That's little different from traveling to any other country really. We had similar experiences in Austria, Russia, Morocco, and Spain for example.