Is it possible to use gRPC with HTTP/1.1 in .NET Core?
No. The RPC call is done only over HTTP/2. This allows gRPC users to automatically leverage all the features of the protocol.
No, you cannot use gRPC on HTTP 1.1; you may be able to use the Grpc.Core Google transport implementation, however, instead of the managed Microsoft bits; this targets .NET Standard 1.5 and .NET Standard 2.0, so should work on .NET Core, and uses an OS-specific unmanaged binary (chttp2) for the transport.
For client-side, there is virtually no difference between the two; only the actual channel creation changes, between:
with the Microsoft transport, and
with the Google transport. All of the rest of the APIs are shared (in Grpc.Core.Api)