If Chris Froome decided that his injuries were severe enough to hold him back from down well in the race, then he needed to pull out. No reason to keep pushing and risk another possible injury or the agitation of his current injuries. He needs to rest his current issues and get ready for the next race. No sense in pushing himself past his limits without a solid chance to perform well.
Chris Froome is a professional athlete who has trained extensively for this event. He knows his one limits, and if in conjunction with his team he decided that it was best not to continue, then that was almost certainly the right decision. It was likely not made lightly and should be respected.
No, Froome should not have continued on with injury. It is my belief that no one should ever continue on in sport with injury for risk of greater injury, especially in the extreme circumstances given by the Tour de France competition. Greater injury would've also led to Froome's inability to participate in further competitions, including the next Tour de France.
A cyclist uses his or her hands and wrists for a lot of their sport. The conditions of the road were wet and the added traction loss would've meant more strain on his already injured wrist. Despite the X-rays showing no damage, he admitted it was painful. For marathon races such as the Tour de France, it is important to take care of one's body first. Had he continued and finished, his injury would've likely worsened. He was wise for sitting it out.