You could do something similar with Bitcoin. It would be slower, but that needn't be a big problem.

If you want to understand Ripple, you have to make sure you understand the difference between the roles of validators, wallets and gateways.

Validators make decisions on which transactions make it into the ledger, much like Bitcoin miners, though there's no equivalent of a block reward. And like bitcoin miners, they don't get to decide the rules that determine which transactions are valid. Unlike bitcoin miners, they do have a vote on some things like fees and minimum reserve requirements.

Wallets are just like bitcoin wallets, they are just collections of private keys and maybe some extra information like labels for otherwise pseudonymous addresses of others. You can host them yourself, or use a hosted wallet service. You do not need to hold any IOUs issued by gateways to use a wallet, though that will limit you to using XRP and any fiat trade credit provided by your extended social network.

Gateways hold fiat currencies and issue and redeem IOUs for them. They need not hold wallets or run validators, though most gateways probably choose to do that as well. They do not keep the books, they merely allow fiat currencies to be traded on the Ripple ledger by honouring their IOUs. This is very similar to the idea of using coloured coins with Bitcoin. The blockchain can be trusted to keep accurate records, but you have to trust the issuers of coloured coins to redeem them for the real world assets they represent.