You can find plenty of Native American restaurants in the southwest. There was one just outside Dallas when I lived there about twelve years ago. I can't recall its name.
But in the broader sense, most of what we consider to be American cuisine is Native American cuisine, just assimilated and adapted in various ways. The Timucua people of Florida used to have a special technique of cooking a goat slowly in a pit dug in the ground with a pot under it to catch the juices. They'd use the juices to make a sort of broth that would be poured over the goat meat before eating. They called it barbacoa. Today we call it barbecue.
My own people are from the Louisiana bayou, and the Indians around those parts are basically the reason why my ancestors didn't die of starvation after the Derangement. It was the Choctaw who first taught my people how to grind up the leaves of the sassafras plant into the powder we now call filé, which we use to make our filé gumbo. (When we don't make our gumbo with filé we make it with okra, and we learned that trick from the Choctaw too.) We had to learn to adapt our French recipes (which were useless in the new and different climate) with help from the Indians and the Caribs and the West Africans who'd been brought over as slaves. We had to learn fast, too, or go hungry.
Course, most of what people think of as "Tex-Mex" cuisine is really Aztec in origin. It was the Aztecs (or maybe the Mayans, I can't remember) who first learned to cook corn with lime (calcium hydroxide, not the citrus fruits) so it could be ground up and made into corn flour and then made into things like tortillas and tamales. If you magic-wanded an Aztec man into a modern Tex-Mex restaurant, he wouldn't recognize everything on the menu, but it would all seem eerily familiar.
So anyway, long story short, there are tons of distinctive Native American foods. We just tend not to notice them all that much because we think of them as American foods.
But ... there are distinct Native American foods.
Fry bread is one that many people will recognise right off. It's good. Often used to hold other food.
Most of the things we do with corn, Native Americans taught us, including cornbread.
Barbecue! If you expand Native American to include the first part of the New World encountered by Columbus, you get this delicious method of cooking meat.
So much of what we think of as "Mexican food" is really "Native American food".
This site has an exhaustive list of books you can read to really appreciate the immensity of the Native American contribution to world cuisine.
If you want to see long lists of things, the Wikipedia article isn't terrible.
I hesitate to suggest this article, as I don't find the argument very convincing, but your question does remind me of a recent piece on the lack of Maori restaurants in New Zealand:
Morris, C. “The Politics of Palatability On the Absence of Maori Restaurants.” Food, Culture and Society: An International Journal of Multidisciplinary Research 13, no. 1 (2010): 5–28.
I think the linkages the author makes between settler colonialism in New Zealand and the palatability of Maori food to a settler audience overreaches, but there are some extended descriptions of attempts at opening Maori restaurants that are interesting. But the author does fail to discuss examples of indigenous food restaurants that other posters mentioned, nor does it deal with the success of Hawaiian restaurants in Hawaii. It might be worth a look anyway, though.
There are a lot of good answers for the impact of Native American cuisine. To shed some light on the restaurant aspect of the OP's question, a good portion has to do with immigration. Many ethnic-themed restaurants correspond to immigrant groups (Mexican, Italian, Chinese) and these restaurants often homogenize a very diverse food culture through their menu. Often times this homogenization overlaps with the immigrant community's origins. For example, a lot of a "typical" US Chinese menu derives from Cantonese or Szechuan cooking, the regions where a the majority of Chinese immigrants came from prior to the 1990s. Moreover, being a restauranteur was a viable and lucrative business model to many of these immigrant groups. For Chinese, owning a restaurant was one of the few ways around the various exclusion acts that prohibited Chinese movement within the US.
Native Americans had a much different experience than other ethnic groups within the US by virtue of being the original inhabitants of the land. The federal government policies of reservations concentrated populations of Native Americans and increasingly segregated them away from the larger metropolitan populations. Metropolitan cities, especially with their heterogeneous culture and milieu, are vital for many ethnic themed restaurants to gain ground among consumers outside the ethnic community. In the words of the sociologist Alan Warde, the metropolitan public sphere (things like magazines, newspaper reviews, daily experience) is vital for "routinization of the exotic." In places like the Southwest, the large Native American population meant that dishes like Navajo Tacos did manage to make the transition into regional dining patterns. Other regions have made similiar patterns of cultural diffusion (as indicated by the other answers).
Sources
Kalb, Don. The Ends of Globalization: Bringing Society Back In. Lanham, Md: Rowman & Littlefield, 2000.
Shortridge, Barbara G., et al. The Taste of American Place A Reader on Regional and Ethnic Foods. Lanham: Rowman & Littlefield Publishers, 1999.
Warde, Alan. Consumption, Food, and Taste Culinary Antinomies and Commodity Culture. London: Sage Publications, 1997.
Some of the food Native Americans ate would not be appetizing to modern Americans. I'm currently reading Comanches: The History of a People by TR Fehrenbach. In this book, the commanches, and plains tribes in general, are described as eating meat raw. The organs were especially relished after a kill, and a special delicacy was the sour contents of the animal's stomach. In addition, they would squeeze out all the contents of the intestines between their teeth. Vegetables were usually foraged and spices are not mentioned at all. They avoided fish and pigs unless hunger forced the issue. This diet agrees with what is stated in Nine Years Among the Indians, 1870-1879: The Story of the Captivity and Life of Herman Lehmann.
i used to live close to a fairly large native reservation in canada and they had a few native american restaurants there. i also worked at a gas station that was on the reserve while i was in college and a few of my co-workers were native, one of which had offered to make lyed corn soup, fried bread, or "indian tacos" for me, which i gather are native american dishes (i don't know with certainty that they are traditional).
i tried to find a website for the turtle island restaurant but the best i found was a community business directory that listed them.
http://www.mbq-tmt.org/community/business-directory?category=5
i apologize if this is not detailed enough for the sub's standards.
There are many distinct Native American foods, most are just not well known. Filé-Crawfish Stew, a food that is sometimes believed to be of Creole or Cajun origin, was originally made by the Choctaw. Pemmican, cornbread, and some variation of jerky are all of Native American origin. I do not have any books with me at the moment, but "Spirit of the Harvest: North American Indian Cooking" by Beverly Cox and Martin Jacobs is a good source of information regarding this topic. I attend a tribal college and am currently enrolled in a course on Native American Cuisine and another on Native American Culture and Traditions.
Salmon n' Bannock is a Native American restaurant in Vancouver, Nek8arre at the Traditional Huron Site in Quebec, Tea-n-Bannock in Toronto are 3 First Nations restaurants I managed to find using Google. There are certainly not an abundance of them but I know I've been to 2 others I can't remember the names of. Not sure this is okay for this subreddit. Sorry if it's not.
The Mitsitam Cafe at the Smithsonian National Museum of the American Indian in Washington, DC serves Native and Native-inspired foods, many crafted from distinctly North American ingredients and techniques. They have a cookbook -- The Mitsitam Cafe Cookbook -- which, along with lots of super delicious recipes, talks about the ingredients and cooking methods various Native tribes use(d).
To add to the other posters' comments about Native American food just being American food, the very first recipe in the cookbook is for guacamole -- which is now so ubiquitous in American cuisine that you can get it on your Subway sandwiches -- and the sidebar notes:
Cultivated in Mexico's Tehuacan Valley as early as 6000 BC, avocados were a popular part of the otherwise low-fat Mesoamerican diet. Thirty percent of an avocado's weight comes from oil, and it is the richest in protein of all fruits. The avocado sauce today called guacamole closely resembles its Aztec precursor, which was known to its inventors as ahuaca-mulli.
The reservation next to my home town sells "Indian Tacos" all the time. Frybread with venison, corn, lettuce, and tomato. So good. Also corn soup, rabbit with corn biscuits and gravy, venison jerky. A lot of deer meat and corn.
This question has been explored in depth, off /r/AskHistorians, but elsewhere on Reddit:
In a nutshell, a large portion of food we currently eat in America is Native American, particularly "Mexican food."
There used to be one in Broadway, CA called The Burning Tree. It closed down not long after 9/11. I would kill for some of their acorn soup.