Start with the actual reason given
Insurers must state a specific ground for rejecting a claim, and the most common ones are a stated pre-existing disease exclusion, an unmet waiting period, an alleged non-disclosure at the time of buying the policy, or a third-party administrator's assessment that treatment wasn't "medically necessary." Each of these can often be challenged if the policy document doesn't clearly and specifically support the insurer's position.
Your first real option: the insurer's own grievance process
Every insurer regulated by IRDAI is required to have an internal grievance mechanism, and escalating there in writing — clearly, with your policy number and the rejection letter — is usually the fastest first step, and it's free.
If that doesn't resolve it
The Insurance Ombudsman offers a free, relatively fast route for individual policyholder complaints up to a specified claim value, and doesn't require a lawyer to use. For larger or more contested claims, a Consumer Disputes Redressal Commission under the Consumer Protection Act, 2019 can award more and handle more complex disputes, though it takes longer than the Ombudsman route.
Timing matters here too
Consumer complaints generally need to be filed within two years of the rejection, and the Ombudsman route has its own, typically shorter, internal timelines — so it's worth acting once the internal grievance process has been exhausted rather than letting it sit.