I'm going to give you the benefit of the doubt and assume you live somewhere that doesn't allow you to walk on water for half the year. The big thing is being able to dry your hand. He's out on the ice with gear, food, equipment, etc. He'll be able to dry off and warm up.

And even if you were stranded out in the wilderness with nothing but your clothes, you could probably get away with getting one hand wet because it's still a relatively small part of your body. Shake off your hand, wipe the excess water on your shirt or pants, and then shove your hand down your pants or under your armpit to warm it up. No big deal.

The real problem is getting large portions of your body and/or clothes wet. Like falling through thin ice. You gotta dry off immediately. Remove wet clothes right away. Roll in the snow. Yes, the snow. It's actually fairly absorbent. And then do what you can to stay warm. Keep your extremities close to your body, get out of the wind, find shelter (even if it's just some bushes or trees), maybe you have the skills to make a small fire, etc. Maybe you can make it through. If you're prepared for the wilderness you have clothes made of materials like wool that keep you warm even when wet. You've got emergency supplies (knife, hatchet, matches, fire source, food rations, etc) and you'll probably be ok.

More rambling: Frost bite tends to be a slow process. You get lost on a hike and spend way more time outside than you planned. Or a storm picks up and pins you down. Either way you're stuck outside overnight or longer. Over time your body naturally reduces blood flow to extremities like fingers and toes, making them more at risk for frost bite. You tend to lose feeling first and may not realize it's happening. Or you may not notice that your feet are wet inside your boots from sweat or melting snow soaking in. Wet makes it worse because water steals heat from your body, either through simply warming up the water or causing the water to evaporate. Anyway so depending on a variety of factors including how cold the weather is, wind, your clothing situation, shelter situation, wetness level, age and general health, etc., you could begin to experience frostbite on parts of your body in a matter of hours or as long as days.

People have survived some pretty incredible situations. This lady from Minnesota, for example, managed to survive a situation in well below freezing temperatures that probably should have killed her.

Anyway, it's currently 25 deg F outside where I'm at, and that's considered pretty warm for this time of year. Hope you're someplace warmer and found this comment interesting.