Of course she can leave it off. Now I'm sure that your friend has her reasons like just not wanting anything from the father or having anything to do w. him but I wonder.
I wonder if she has really thought this through. Being a single Mom in the military, while doable is going to be tough. And we all know what military pay amounts to. If the father is military it could actually be better career wise to name him as the father.
Regardless the day may come where a bit of child support could make a really big difference in the welfare and future of the child. Now she may just not want him to have any rights if she does this. But even if 10 years go by he could still come around and sue for them even if not on the certificate. It happens.
If named as the father she could tell the guy that she won't go for support as long as she never sees him again. Or get him to sign full and total custody over in exchange for not exercising the right for support AT THIS TIME. With that over his head he may never chose to come around in ten years.
So though I don't really know something vaguely itches at the back of my mind that just leaving him off could be troublesome later. She should really speak to JAG (especially if he is military) and explore the possibilities. If she is uncomfortable with JAG (some are just awesome and some are ho hum) she could talk to a civilian attorney.
My instinct is there will be no harm putting his name on the certificate. If she never sees him or he does not know what is the difference? If something happens later she could pul the "well you now owe me $1000 a month" card.
I'm also thinking that later, say 25 years from now, the kid may look up her father..... people have a burning desire to know their parents. So Mom finally explains that dad was a jerk, it was one night or whatnot. Dad explains that if he only knew (hes not on the certificate) he would have been a great Dad and been there in some way. If the name is on the certificate Mom can say that he chose to not be involved.
Pardon my ramble..... just something to think about.