Yes, his credibility has suffered and he faces a difficult time in the next couple of months. If England lose their opening two World Cup qualifiers I could see his opponents building a worthwhile case against him. But, for now, the case for sacking him is flimsy, especially after the clumsy attempt to offer him as a sacrificial lamb to the News of the World.
Result: Stay. As long as he gets his tactics right I expect him to bounce back with a win.