Upvote:10
You can and you are expected to turn into the nearest lane. Whether there is traffic in the other lane is apparently not relevant.
The Washington State Driver Guide says:
Turn from the lane that is closest to the direction you want to go and turn into the lane closest to the one you came from. When making turns, go from one lane to the other as directly as possible without crossing lane lines or interfering with traffic. Once you have completed your turn, you can change to another lane if you need to.
That said, it's a very good idea to wait if you aren't absolutely sure that the driver of that vehicle isn't going to suddenly change lanes without signaling. Countless dashcam accident videos on YouTube taught me this... I'd go only if I saw him move from the right to the left lane to let me in.