Over 600,000 newborns in India died in their first month of life in 2017 despite large increases in access to maternal health services. We assess whether maternal and newborn health system quality in India is adequate for institutional delivery to reduce neonatal mortality. We identified recent births from the cross-sectional 2015–2016 National Family Health Survey and used reported content of antenatal care and immediate postpartum care averaged at the district level to characterize health system quality for maternity and newborn services. We used random effect logistic models to assess the relationship between institutional delivery and neonatal (death within the first 28 days of life) and early neonatal (death within 7 days of live births) mortality by quintile of district maternal and newborn health system quality. Three quarters of 191,963 births were in health facilities; 2% of newborns died within 28 days. District-level quality scores ranged from 40 to 90% of expected interventions. Institutional delivery was not protective against newborn mortality in the districts with poorest health system quality, but was associated with decreased mortality in districts with higher quality. Predicted neonatal mortality in the highest quintile of quality would be 0.018 (95% CI 0.010, 0.026) for home delivery and 0.010 (0.007, 0.013) for institutional delivery. Measurement of quality is limited by lack of data on quality of acute and referral care. Institutional delivery is associated with meaningful survival gains where quality of maternity services is higher. Addressing health system quality is an essential element of achieving the promise of increased access to maternal health services.