With the proviso that answering this for two continents, many hundreds of tribes/nations/groups, and over 18,000 years would be pure folly, I can tell you that from my perspective in the southern Northwest Coast (NWC), the answer is "not exactly."

Like virtually all other aspects of NWC culture, childbirth was rich with rituals and cultural conventions. There were dietary rules, residential rules and rules about travel for pregnant women. Special formulae of song, dance and medicine were used to assist with labor and other aspects of the actual birthing process. Activities around birth (e.g. naming conventions) were heavily constrained by rule and ritual. These kinds of norms surrounding birth are, in my experience, common among Native Americans and are just as likely to be noted for the Cherokee or the Powhatan as they are for NWC groups.

Given all these rituals and rules, it is not particularly surprising that there were special women that assisted in many of these women's practices. These women were usually from the extended family - often mothers, grandmothers, great aunts, or other relatives depending on kinship rules or a variety of other factors. Elderly, knowledgable women were respected and played an important role in the extended family/settlement. They conducted coming-of-age rituals for women. They carried and handed down spiritually powerful songs and dance formulae. They acted as advisors in matters of the family, marriage and decorum. And they played an important role in the preparation for, actual childbirth and follow-on related cultural practices.

There were women that assisted in childbirth, but I don't see that role as a mid-wife. It was not an occupation or a professional service. It was more like a family role, in my view. If medical help was needed in childbirth, it would be provided by a doctor, and that would be analogous to a midwife in that it was a service provided and remunerated.

Menstruation and the Power of Yurok Women: Methods in Cultural Reconstruction.Thomas Buckley American Ethnologist Vol. 9, No. 1 (Feb., 1982)

Life and Culture of the Hupa. P.E. Goddard UCPAAE Volume 1, 1903

Yurok Marriages T. T. Waterman, A. L. Kroeber UCPAAE Volume 35, No. 1, 1934

Perdue, Theda, Cherokee Women: Gender and Culture Change, 1700-1835. Lincoln: University of Nebraska Press, 1998.

Rountree, Helen. Powhatan Indian Women: The People Capt. John Smith Barely Saw. Ethnohistory, 45 (1998) 1-29.

Edit: Fixed references