As far as I can tell, the person who actually leads the Gates Foundation health division is Trevor Mundel. Even then, the work of the Gates Foundation has been in funding projects lead by other organisations, rather than in implementing those projects themselves.
If they do it very well and can point to measurable results of success, I would think so!
His work in eradicating Polio. Choosing which projects to finance is a skill in itself, something he's likely more capable of than a traditionally credentialed expert. I can't help but think that you have an arbitrarily narrow definition of expertise.
There is no basis at all for claiming he’s a public health expert. He might be an effective philanthropist, but that’s a completely different topic.