Native Health and Wellness Act of 2017 This bill amends the Public Health Service Act to require the Centers for Disease Control and Prevention to award grants to Indian tribes or tribal organizations for promoting health, preventing disease, and reducing health disparities among American Indians and Alaska Natives. Specifically, grants may be used to: establish or support preventive health service programs that facilitate the achievement of health-status goals; establish or support public health services that reduce the prevalence of chronic disease among American Indians and Alaska Natives, or strengthen public health infrastructure to facilitate the surveillance and response to infectious disease and foodborne illness outbreaks. In addition, the Department of Health and Human Services must make grants to Indian tribes and tribal organizations for recruiting and mentoring American Indian and Alaska Native youth and young adults in health professions.