There won't be anything stored on the watch by runkeeper. If you start a run from your phone, does it show up as a card in your watch? That's all Runkeeper does with Android Wear on top of add the voice commands. Does Track a Run work with Runkeeper with your phone and not the watch? If not, that's probably the problem. If it does, you might need to initiate the first one from your phone while the watch is on you and then it might recognise it after.
EDIT: Is that watch a Sony Smartwatch? If so, apparently the app does show up there as you can run without your phone being present according to that page. You can't with other watches.