After 2 1/2 months of Apple Watch ownership and frequent use, the touch actions had become either erratic or irresponsive. Scrolling via swipe sometimes worked and pressing a button or notification rarely had an effect. As the watch is really a computer, I should have realized earlier that it needed to be rebooted.
The fix is simple: perform a hard reset on the watch. To do this, simultaneously push the digital crown and sleep button until the Apple logo appears. This seems to have fixed the watch (and it seems to be a little faster, too).