No, they're two different systems. The ombudsman advocates for an individual resident; the Indiana Department of Health investigates regulatory complaints and can take enforcement action against a facility's license. Using one does not preclude using the other.
How this actually works
Filing with both, when warranted, is common and appropriate -- they answer different questions and produce different kinds of outcomes. The ombudsman can help resolve a specific resident's situation quickly, sometimes within days, while an IDOH licensing complaint triggers a formal, slower investigative process that can affect a facility's standing broadly, not just for one resident. If you are unsure which applies to your situation, calling the ombudsman first is a reasonable starting point -- they can point you to IDOH if the issue is regulatory rather than personal.