Delete records in Peer to Peer Replication in SQL Server 2008 R2
Peer to Peer is transactional. So the log reader will re-generate the result of the delete, row by row based on the transaction log. The there will be 8+ million delete operations performed on each of the other nodes. And, yes, you only run the procedure on one node. But as you probably understand, having row-by-row as the end result is far from optimal.

An option is to include the procedure in your publication. This mean that SL server will, for you, execute that procedure on the other nodes (instead of sniffing the log).

Above is for transaction replication. I can't say how peer to peer changes things, if they do. Make sure to test thoroughly.

https://docs.microsoft.com/en-us/sql/relational-databases/replication/transactional/publishing-stored-procedure-execution-in-transactional-replication?view=sql-server-2017