Avoiding Joint Injuries
Small joints like those in the fingers are designed for dexterity and manipulation, not to support your full body weight. In this (image) you can see just how small the distal joint actually is. Protecting these small joints for the long term for all the things your hands need to do is better than using them to do pushups - imo.
Increasing the Size of your Wrists
Size of the wrists and strength of the wrists are not necessarily the same. The size of your wrist joint is primarily dependent on the size of your bones, not muscles. The muscles that control the wrist are forearm muscles. Exercises specific to the wrist and forearm can increase both bone and muscle growth.
Increasing the Strength of your Wrists
If your wrist strength is the weak link in your biceps curls or other arm exercises, strengthening your forearm muscles will help to strengthen and protect the wrist joint. Isolating the wrist with exercises that target the wrist flexors/extensor, radial/ulnar derviators and pronator/supinator muscles will increase wrist strength by working the muscles with tendons that cross the wrist joint. These exercises will specifically allow you to address your weakest link:
As far as increasing the bone growth (size and density), bones respond to stress loads with weight bearing or resistance exercise. Assuming you have adequate nutrition including calcium and vitamin D, by gradually increasing the amount of stress to your wrist bones with increasing weight bearing or resistance loads the small wrist and arm bones will adapt by increasing bone density. Body weight exercises such as Yoga's downward dog apply body weight thru your arms and wrists with your wrist in good alignment.