Technically possible? Absolutely: that 30A outlet has two 120V legs, neutral, and earth-ground. So yes, technically possible.
A Good Idea(tm) though? NO. Absolutely not.
The breaker and wiring for the dryer circuit are for 30A. The 120V outlet for the washer will be a 15A (or perhaps 20A) device -- so the available current (30A) would exceed that device's safety rating.
Various electical codes also forbid it -- the washer and dryer, like most permanent appliances, are typically each required to have their own, dedicated circuits.
Cheers
Thanks for the info!
That's mostly what I had thought. I'll be getting it all wired up the correct way! On that note the outlet is currently a three prong type, I've read on-line (wikipedia!) that it's now considered an obsolete type and replaced by a four prong design. Is it worth it to have that swapped out as well? The installers helpfully left behind that type of cord.