We are trying to run the .net core API service on Cent OS using Nginx server. Do we need .NET SDK or run time libraries are enough to run the .NET service on Linux?
It depends on how you are publishing the application.
If your application is published as a Framework Dependent application, it will need just the .NET Runtime to be installed.
If your application is published as a Self-Contained application, it will not need the .NET Runtime to be installed (but the transitive native dependencies of .NET runtime - such as OpenSSL and ICU might be needed).
At no point should the SDK be required to run the application. If it is, you are doing something wrong, or have run into a bug.
For more details, see:
If you are targeting Linux running Intel x86_64 machines, remember to use the linux-x64 Runtime Identifier when targeting the application.