Yes, FMLA is for fathers as well.
Technically, they can't discriminate or hold it against him if he uses it. However, I know a gentleman at DHs work who tends to abuse his sick time, time off, etc. he plans on taking numerous weeks of FMLA and although they can't discriminate, they are very much aware that he is once again trying to get the max out of his time off. There are four other men in the office whose wives had or will be having babies and they are all taking the typical 1 to 2 weeks.
For the record, the law only says that the company has to give them the unpaid leave and continue their benefit coverage. There is nothing about holding it against them. If a promotion opportunity comes up in a year, and the boss says "Jim took 12 weeks when his kid was born, Bob took 2. Looks like Bob is more committed to the job and is a more needed part of the team - he gets it", they have every right to do that. So no, they can't fire you, but it can certainly affect your H's career.
My H has worked hard to make sure he is indispensable to his boss and team. Taking more than a week or so off would give the boss a reason to say "hmmm, we're managing pretty well without Joe...". Not how you want to be seen in an economy like this. So he will be back at it within a week or so, depending on what day of the week this blessed event happens.
ETA: when we took 2.5 weeks for wedding/honeymoon 3 years ago, his boss said "we need you. Please don't do that to me again!". If your job can be easily done without y for 12 weeks , that's not good...