You can keep your scores and stats right from your Apple Watch when you Start a Round in SCORING mode (previously called Simple Input) on your iPhone. This is a Premium feature, please head to this article to see how to use Hole19 with your Apple Watch.
If you are not able to swipe on the Distances screen of your Apple Watch, you probably started a round using SHOT-BY-SHOT (was Advanced Input) which is designed to let you track your round shot by shot on the iPhone. It's also possible that you're starting a round selecting 'GPS Only'. Make sure that you're choosing the correct input mode before starting your round.