Family Law Questions? Ask a Family Lawyer Online.
Either parent can initiate the proceeding to terminate, but it's not likely to go through unless ex-wife's new spouse is willing to adopt the children as his own (step-parent adoption). The step parent adoption process is not as arduous as a non-parent adoption, but it's still a court proceeding, and the court must find that it's in the best interest of the children to be adopted by the step parent.
Judges must consider the best interest of the children here. A Judge might consider that children that have already been adopted should not have to go through the process again. So it doesn't necessarily matter that the exwife thinks it's in the children's best interest, the Judge has to agree with her. Your son doesn't have to give up his rights, and frankly, ex-wife probably has very little grounds to insist that he do so. Most terminations have to either be voluntary or there must be some compelling reason to terminate the dad's relationship (drugs, crime, abuse, etc).