All sorts of entirely credible, predominantly bricks-and-mortar schools offer the MPH by distance learning with many with CEPH accreditation, the MBA or MMM by distance learning, including I think every ACPE-approved program designed specifically for physician executives, master's degrees in medical education, health informatics, epidemiology, patient safety, pharmaceutical regulation, etc., etc. My family physician, who held a senior academic post, has a postgraduate Master's of Clinical Science in Family Medicine from the University of Western Ontario. It may have been part-time on-the-ground at the time but today it's delivered by distance learning.
Now there's much more activity at the master's than the doctoral level, but there are still unimpeachable, substantial, usually bricks-and-mortar schools with relevant doctorates by distance. The DPA from Valdosta State University. The DSc in information systems with healthcare IS emphasis from Dakota State University. The DHA from Central Michigan University. These are public universities. The DHSc, DBA, DPA or EdD from Nova Southeastern University or the DHSc and DHEd from A.T. Still University; these are non-profit universities which also house accredited schools of osteopathic medicine, dentistry, and others. Nova is the largest private university in Florida, ATSU the founding school of osteopathic medicine.
Whether self-funding a hugely expensive degree - in public policy and administration! - from what I'm guessing is a mostly- or entirely-distance learning school, probably for-profit, probably with a very small academic footprint, if any, is the right thing to do for the OP is one question.
But more generally, there absolutely are doctorates, numerous relevant to medicine, delivered wholly or in large part by distance, that are as rigorous and credible as doctorates delivered in larger part more traditionally.
