Hasan Savran shows how you can set unique key constraints on Cosmos DB containers:
Unique key names are case-sensitive, I have good experience on this. If your unique key is in lowercase letters but your data has field with uppercase, CosmosDB will insert null value into unique key first time, you will get an error second time when it tries to insert null again. CosmosDB does not support sparse unique keys. If your unique key is /SSN, you can have only one null value in this field.
If you like to use unique keys in Azure CosmosDB, you have to them when you create your containers. You cannot add a unique key to an existing container. Only way to add a unique key to an existing container is, to create a new container and move your data from older container to the new one. Also, you cannot update unique keys just like partition keys. Picking a wrong unique key can be an expensive error.
Looks like you’ll need to have a bit of foresight when choosing keys (or choosing not to use keys).