This is hard to answer due to the relative sparsity of historical records for many regions, but we do have some information for contact era cultures especially the Aztecs. I'm also going to ping /u/400-Rabbits since he's more familiar with the Aztec records than I am.

Many cultures appear to have had specialist healers and midwives which made use of a wide array of medicines made from both plants and minerals. The Relaciones Geograficas de las Dioceses de Michoacan 1579-1580 lists, among other things, almost 100 medicinal plants and their uses. Sahagun (a Spanish chronicler of the Aztecs) has a similar list. To give you a short selection from the former source, a colonial administrator from Cuitzeo (A Tarascan community from Northeast Michoacan, or West-Northwest from Mexico City) lists the following plants as common medical treatments in his region:

• Andámucua - An herb that looks like starflower which was first chewed and then held in the corner of the mouth. The herb produced heat and acted as a local and general anesthetic which was often used to treat toothaches, etc. It also had effects similar to caffeine and was popular among workers as a way to gain an extra boost of energy during the day.

• Chupírini - This tree was cut to harvest sap which was then mixed with water or broth and drunk. It was used as an anti-inflammatory.

• Janbánducua - This was a root which was used in the treatment of broken bones, although the source does not specify exactly what it did.

• Pejorita - An herb which was also used in bone injuries, possibly as an anesthetic.

Other medical treatments are described for treating stomach sickness and for use as a disinfectant for cuts. Pulque (an alcoholic beverage made from fermented agave nectar) was also believed to have medicinal properties and was often used as a base for other medicines. More complex medical treatments, like trepanation, were rare. Other than plant and mineral medicines, the use of sweatbaths or hot springs were commonly thought to hold medicinal value. (One rather racist administrator in the Relaciones Geograficas writes at length about how the Indians in his town bathe too much in the local hot springs, and he argues this is what is making them so susceptible to smallpox and other Early Colonial maladies.)

Childbirth is also discussed at length by Sahagun. Pregnant women would begin consulting a midwife early on, and as the pregnancy neared completion they would spend lots of time in a sweatbath (temazcal). Often the midwife would massage the pregnant woman's belly to try to position the infant correctly before birth. During the birth itself, the woman would be attended both by the midwife and female relatives. In the event of complications, the midwife could administer medicine that induced contractions early.

As a side note, child birth was seen by the Aztecs (and possibly other Mesoamerican cultures) as the female equivalent of battle. Deities of childbirth and fertility (e.g., Itzpapalotl) were also associated with death, and bringing a child in the world was often equated with taking a captive in battle. Aztec women who died in childbirth were given warrior funerals.

I was unable to find any evidence that elites had access to medical treatments unavailable to commoners. However, a bioarchaeological review of health and nutrition does indicate that elites were overall much healthier than commoners (Steckel 2005). In general, urban populations in Mesoamerica had lower overall health than other Native American populations, but with elites having less frequent incidence of disease or other maladies. Steckel (2005) attributes this difference not to differential access to medical treatment but to better nutrition.

As I stated before, a lot of the specifics are difficult to assess. Medical treatments don't often leave archaeological evidence, so we are forced to rely on often meager historical sources.

Sources:

• Evans, Susan Toby. 2008. Ancient Mexico and Central America: Archaeology and Culture History

• Garciakutzbach, A. 1976. Medicine Among the Ancient Maya. Southern Medical Journal. 69 (7). pp. 938-940.

• Relaciones Geograficas de la Diocesis de Michoacan 1579-1580. Papeles de la Nueva España. Colección Siglo XVI. Ernesto Ramos (ed.) 1958. Guadalajara, México.

• Sahagun, Fray Bernardino de. 1969 [1569]. Book 6 of the Florentine Codex: Rhetoric and Moral Philosophy.

• Steckel, Richard H. 2005. Health and Nutrition in Pre-Coumbian America: the Skeletal Evidence. Journal of Interdisciplinary History. 36(1)