Unfortunately, you cannot. If you already have a verified coinbase account with a bank account attached to it then i would really just recommend you buy bitcoin/ethereum first(i would recommend ethereum for small amounts, since transaction fees are somewhat lower), and then sell them on either bittrex or shapeshift.

Another option would be trading p2p, of course. But depending on the specific altcoin you are looking to buy there might be an issue finding sellers that would accept your fiat for it directly. There are certain exchanges that have fiat and altcoin pairs but usually your choices for altcoins are quite limited, and you'll have to go through bitcoin anyways.