As your main reason to shy tipping is ‘being uncomfortable doing it‘, I’ve added this answer, even though it misses the question given.

In sit-down restaurants, normally nobody sees you doing the tipping - it happens after you paid, and before you leave. Staff will only know the amount of your tip when you have long left (from experience, this is not known to many people outside the US).

The usual sequence in the US is:

1. You get the check (bill), server walks away.
2. You inspect it at your own time, nobody is watching.
3. (If you agree with it,) You put your credit card on it
4. Server picks it up, walks away to the cash register with your card, and brings it back for your signature, with the full amount charged to your credit card. No tip is added or mentioned at this time.* Server walks away.
5. You take your sweet time deciding on tip amount, add it, and sign. The amount could well be zero, if service was really bad (but that is rather harsh). Then you get up and leave.
6. Someone comes and cleans the table, and takes the signed check. Later in the day, someone types the new total (including the tip) in, and your credit card charge gets adjusted.

This removes the need to interact with the server about any piece of the tipping process. Maybe that information removes your discomfort.

* Some restaurants charge a mandatory 18-20% "gratuity" for parties over a certain size, most commonly six or more people

Since no-tipping restaurants are very rare in the U.S., your best bet is going to be to find restaurants where you are doing much of the work for yourself - counter-service restaurants, for example.

There are many such restaurants in the U.S., although they veer toward the casual (most fast food restaurants are such). Fried chicken and barbecue restaurants very often have counter service. A good seafood joint in San Diego with counter service is Point Loma Seafoods, as another example. Yelp should be able to help you find such restaurants.

Another strategy? Just add 20% to the prices you see in the menus, and consider that to be the price of your meal. If you'd eat there if they printed that as their price, in effect, that's what you're paying with a 20% tip (which is a slightly generous tip in the U.S.).

A final strategy? You can completely avoid the tipping issue by renting hotel rooms that have kitchenettes or full kitchens, and preparing your own meals. You'll save a lot of money too!

I am uncomfortable with tipping and would much rather eat at sit-down restaurants in the United States that pay their employees fairly without expecting customers to supplement with tips (as occurs in many other countries in the world). In other words, I want to eat at restaurants that explicitly do not allow tips.

You can't.

That's simply not how it works in the US. The restaurant in the news article you linked is notable because it is extremely rare (like, a tiny tiny handful across the entire US).

If you are uncomfortable with tipping in the US, then you need to choose fast food or counter-service restaurants, period.

At table-service restaurants in the US, tipping is expected. Is it the law? No, but this is an extremely strong norm in the US.

Let me emphasize this point, which has been much discussed in comments: failing to tip at a table-service restaurant in the US would be viewed by Americans as extremely rude.

And not just by the wait staff. On a date? Good luck getting a second. Trying to impress that American business client? Sorry, the other supplier is looking like a better fit.

Is it illegal to not tip your server? No. It's not illegal to write "F--- YOU" on your bill, either.

In general, you don't.

There are exceedingly few restaurants in the U.S. where a tip is not allowed. In most parts of the country, you won't find a single one. There are some where tipping is not expected, but these will primarily only be counter-service restaurants (i.e. fast food,) not table-service restaurants with servers. Even the fast-food restaurants will generally allow tipping, even though it isn't expected. Some will even have a tip jar, though most people still don't tip in such restaurants.

Almost all table-service restaurants where your food is served to you expect you to tip the server in the U.S. Most towns will not even have a single table-service restaurant where tipping is not expected, let alone one where it isn't allowed.

If you really don't want to tip at a restaurant in the U.S., then you should plan on only eating fast food or food that you purchase at a grocery store and prepare yourself.

As far as the bit about payment, though, tips are usually pretty generous in the U.S. People who work at restaurants with tipping usually make significantly more than people who work at restaurants where tipping is not expected. Do not assume that workers are not being treated fairly because of the tipping system. Reality is closer to the opposite of that.

Direct answer: Limit yourself to less formal restaurants, which operate in "cafeteria style", where you go to a counter to order, and are given a pager to come collect your food, or a tent card to put on your table and a runner delivers it. (that's the last you'll see of the runner). These are becoming surprisingly good, that is to say, the food is good. The demographic is middle-to-upper-class, who want an upscale taste without the time and expense of waitservice. Panera Bread, Noodles & Company, Chipotle, Qdoba, Mod Pizza, Smashburger, various Korean BBQ, etc. As well as a variety of locally-owned one-off or mini-chains (Buckhorn, Firewood Grill).

Not to be confused with waitstaffed, must-tip restaurants going downscale by offering plain cheap food but retaining waitservice formalities (and thus the obligation to tip): Denny's, Steak & Shake, Cracker Barrel, your local greasy spoon diner, etc. These places are for people who want to be pampered on a budget, but want plain food that appeals to their palate. In these places you will be seated, a waiter will give you a menu and listen to you give your order verbally, and that means you need to tip!

You can also consider plain old fast-food, like McDonalds, Taco Bell, Wendy's, Burger King, KFC, Arby's, Subway, Sonic, Hardees/Carls Jr, Tim Hortons, Whataburger, Del Taco, White Castle, most pizza places, etc. Their hallmarks are lab-engineered food design, industrial food sourcing and distribution, lousy ambience, salt-of-the-earth clientele, cheap pop music (Muzak), frazzled minimum-wage staff, etc. To get the genuine American experience, make sure to be one of the 70% of customers who orders through the drive-through.

I thought the upscale-cafeteria type, such as Panera Bread, was referred to as "fast casual". But Google thinks otherwise, a search for "fast casual" will turn up 95% "plain old fast food", so that's useless.

If you genuinely have a medical need that requires you get waitservice, focus on fast-casual restaurants and request the staff help you. There is a disability-support law called ADA which obliges them to do what is easy.

Other than that, if you're a waitservice snob who insists on the formalities but can't tip, that is incompatible with being on this continent. That thing doesn't exist because the culture here is so strong, and you can't flip that culture, it's certainly been tried. I am not here to troubleshoot your issue with tipping but to say it may be troubleshootable, feel free to ask a separate question.

would much rather eat at sit-down restaurants in the United States that pay their employees fairly without expecting customers to supplement with tips

This might be a bit counter-intuitive but many restaurants in the States also have something called: service included.

When you get the bill the tip is effectively automatically added. Meaning that you don't need to tip on top of that, but the service fee is already included in your check.

It seems some even market their products this way; this trip advisor question mentions:

they list the tasting menu at $275 service included

You might also see the wording:

Gratuity included

This way you don't need to tip, you just need to pay your bill.

It is common in some high end restaurants to implement a service charge instead of a tip. One such example is Renee Erickson's restaurants in Seattle. Here is an article where she discusses this. Applying a service charge is illegal in some places for parties smaller than 6. In fact, here is an article about a lawsuit over such a practice.

I suggest looking for restaurants with a service charge on Google.