Full disclosure: I haven't RTFA, so I don't know who the author thinks will nuke who. However, the responses here mostly assume it would be a nuclear power nuking another nuclear power. As many have pointed out, having precision nukes would not cause that to happen; it's just too risky.
However, I think that precision nukes do increase the chance of a nuclear power nuking a _non_nuclear power. Granted, I don't think the risk is that high, but there are some possible scenarios where a precision nuke could be used -- maybe a major terrorist attack on the US (lead by a hawkish president) by a group based in some remote area. I'm sure other scenarios could give other nuclear powers an itchy trigger finger too. Again, I'm not saying it will happen, but it's more likely with precision nukes than without.