There's a little ambiguity around the term "full node" here. In order to be an Eth2 validator, you require the following software components:
An eth1 node (geth, parity-ethereum, etc).
An eth2 "beacon node".
An eth1 "validator client" (if it doesn't happen to be included in your beacon node).
For (1), a fast-synced Geth node suffices for now. All eth2 ...