Bitcoin core and Bitcoin protocol

[Marleydee]

Bitcoin core helps to keep it decentralized and is programmed to decide which block chain contains valid transaction
Bitcoin core was created by Satoshi Nakamoto in the late 2008 and is the original implementation of Bitcoin.
  It is invariably a wallet for storing bitcoins and the node of the Bitcoin network. The wallet allows for storage, sending and receiving of Bitcoin.

Bitcoin protocol  https://www.researchgate.net/figure/Example-of-the-bitcoin-protocol_fig1_299368011
 Can Bitcoin core operate without the protocol or do they work hand in hand.

[Pmalek]

Can Bitcoin core operate without the protocol or do they work hand in hand.

I assume this is meant to be a question.
Bitcoin Core is a software. Without the bitcoins on the blockchain, the software would have no purpose or reason for existence. Nodes are essential to the proper functioning of the network because they validate all transactions and make sure that consensus rules are being followed when new blocks are added to the blockchain. They share and propagate transactions with their peers and add them to their mempools. Bitcoin wouldn't be decentralized if it wasn't for the thousands of volunteers who provide a free service - running their own nodes. 

[Zilon]

Bitcoin core depends on bitcoin protocol for its peer-2-peer decentralized transaction. Just like the Http is the protocol to the web. Bitcoin has it's own protocol which operates using cryptographic algorithm. Be aware that the bitcoin core is the refrence implementatin of the bitcoin system. By implementation it includes confirmation of transaction, wallet creation and all enabled by the bitcoin protocol

[FatFork]

Can Bitcoin core operate without the protocol or do they work hand in hand.

No, Bitcoin core cannot operate without the Bitcoin protocol. Bitcoin core provides the application layer that allows users to send, receive, and transact within the Bitcoin network. However, every transaction requires consensus from all network nodes to validate the transaction's integrity. Therefore, for every transaction that is submitted to the Bitcoin blockchain, Bitcoin protocol must be used as well.

[Edith1994]

Bitcoin core is a wallet where Bitcoin is being stored therefore the wallet allow you to store, receive and send Bitcoin.
                                    While
Bitcoin protocol is the stages involve in Bitcoin network for making payment operation.

In my own understanding both Bitcoin core and Bitcoin protocol works together in the sense that one is the wallet (store, send and receive) while one the other hand is the stages involve in Bitcoin network.