How can I find out if a transaction used a specific vout as its input? I want to know if a certain output is spent. I am not using bitcoind, I do not have a gettxout RPC call available. How can I learn if a vout is referenced as an input somewhere in the blockchain or accepted mempool?

I don't mean just for my wallet, I want to know if an output has been spent on the network somewhere.
You need a way of querying the unspent pool in order to do this. A workaround is to lookup the transaction that is being referenced via some public api, blockchain etc, and check the referenced output as they usually mark them as spent/unspent. You can do this programatically by querying for all unspents for the addresses referenced by the output in question and filtering it for the one you want.