To offline any region from the cosmos db account , we need to remove the region from the replicate data globally blade. Please find the below documentation that explains regarding the process to add/remove the regions.
About the API:
This API is managed by product team internally and this shouldn’t be invoked by customers.
1) Cosmos DB automatically triggers failovers of accounts when you enable “Automatic Failover”. The current write region is marked offline and the other regions are promoted as the write region (Based on how you configure them).
2) Automatic failover is part of the DRI and is nearly immediate. SDK is multi-homing enabled and will failover to right region in your priority. You can test how this works with controlled manual failover today. Please refer to : High Avail
Please let me know of any concerns.