In most cases in which babies are abandoned or given away, the mother is unwed and the father may not even know that he has a baby, and typically doesn't care even if he does know.  If he did care, he would probably have married the baby's mother, or at least would be living with the mother and would be sufficiently supportive that the mother would not find it necessary to get rid of her baby.  It is precisely the frequently dire problems associated with being a single parent, that drive women to abandon or give up their babies.  However, if a woman is actually married and then decides that she doesn't want her baby, the father would have a legal and moral right to keep the baby even if the mother doesn't want it.  When fathers accept the responsibilities of fatherhood, they then also gain the rights of fatherhood.  In the case of mothers, when they give birth they automatically have a baby on their hands, whether they wanted one or not.  That's the difference.

Personally I think that there is a better solution to unwanted babies than to make it easier for their mothers to secretly abandon them.  Secrecy is not really socially beneficial.  The secret disposal of a baby raises questions.  What happened to the baby?  Was it murdered, kidnapped, lost?  Do we need a missing persons alert?  Was the baby left at a safe collection site, or merely dumped in a garbage can?

A mother wishing to unburden herself of an unwanted baby should give up that baby openly rather than secretly, by simply taking it to an adoption center and turning it over, or by simply leaving it at the hospital after giving birth.  Maternity wards should offer that option.  Why even take the baby home if you don't want it?  (Although, of course, not everybody gives birth in a hospital, so this route is not available to everybody.)  

Strangely enough, there are large numbers of childless couples who eagerly want to adopt babies, and in many cases spend large amounts of money to acquire babies illegally.  If we make it easier for mothers to hand over unwanted babies and also easier for people to adopt those babies, everybody benefits.  Of course, some adoptive parents do a bad job, but then, some biological parents do a bad job.  That is the risk of parenthood.  As a society we can continue to be vigilant so that if there are abuses they can be corrected.  Nonetheless, we should give people the chance to adopt children, and presume them to be innocent of bad parenting unless proved guilty.    

   I don't know that it's so much that women have all the rights- though the courts often rule that way in custody cases- but that there's a long cultural acceptance for men leaving their families, one that's gone on for centuries. For a lighter example, when Bruce Springsteen wrote in "Hungry Heart":

    Got a wife and kids in Baltimore, Jack,
    I went out for a ride and I never came back

if anyone listening even thought of the lyric, the first reaction wouldn't be, "wow, what a monster", which it might well be if a woman had written that. For a more serious example, think of the Promise Keepers, who made a point of applying public pressure to themselves to stick around and support their families (in part; I know there's more to the group than that); the unspoken assumption is that these men weren't really getting pressure from anywhere else in their lives to do so (except their wives, of course.) Fathers sticking around and the mothers leaving has, until recently, been the exception that proved the rule.
    As for your Devil's advocate question, it would probably end up being neutral on the books and biased when it came to enforcement, in the same way that VW vans get pulled over more than other cars. The "safe haven" laws look pretty gender neutral, at least according to the Wiki, but with a 72-hour window, the chances are pretty good that the mother and child would be in the hospital for at least a third of that time, seriously crimping the opportunity for the father. Getting the local police to go after a mother who had given up a child over the father's wishes could happen, but I could see it being less likely if there was an election to win or lose later on.

Ex ignorantia ad sapientiam; e luce ad tenebras

Does this ENCOURAGE mothers and fathers to abandon their kids?  Maybe.

I'll go half a step further (and likely off the ledge).  Any would-be parents that really want to drop off their baby in a nice safe baby hatch probably should.

Humorless. Cretinous. What'd you expect?

That's because it is a microwave oven.  Hospital baby food... for babies, from babies.

[It's official, I'm going to Hell for that joke.]

Humorless. Cretinous. What'd you expect?

I wasn't talking about the ugly photographer in the background. I meant the man wearing a dress and string of pearls, pretending to be a woman.

I see an ugly woman...you see a guy in a dress.  To each his own...you'll get no judgment from me.

I can't argue with your logic...but I can recommend a good therapist

They need to build one big enough for teenagers. That'll keep 'em in check. Stop playing warcraft or it's off to the baby drop!

make it rain you nappy headed ho's