I really do not even think marriage is a right. Marriage is a responsibility. It’s not a gift that somebody says, ‘Hey, now it’s time for you to get married. It’s our bestowal to you.’ It’s a commitment that you make and it is a responsibility that you accept.Attaturk believes the clammy hand of hypocrisy is stroking Limbaugh's back here, but I'm not sure I'd agree; certainly I would advise caution about implying three failed marriages is a priori evidence of irresponsibility.
My issue with this is somewhat different, and since Attaturk has brought the quote back up and since it feeds into some of my larger issues with the American (and British) Right and discourse in general, it's worth taking a moment to consider Limbaugh's statement in a wider context.
One of the most constant frustrations when engaging with many right-wingers is how totally they fail to grasp the notion that rights carry with them attendant responsibilities. The right to free speech carries with it the responsibility to not call for the extermination of Muslims, or tell people that a healthcare bill designed to save money and save lives is actually going to kill people's grandparents. The right to bear arms (not that I believe such a thing exists, though the difference between legal rights and human rights are probably best left alone for the moment) carries with it the responsibility to ensure guns are kept in safe places, are well-maintained, and aren't waved around at protests as some kind of bullshit display of force and rebelliousness. Both the greatest strength of and the greatest problem with rights is that we recognise that one retains them even whilst ignoring those responsibilities (up to a point where such violation of responsibility becomes literally criminal).
People like Limbaugh don't get that. They see every suggestion that they take those responsibilities more seriously as an attack on their rights (whether this is due to their paranoid victim syndrome, or whether said syndrome is based on this misconception, I don't know; perhaps they feed into each other). "If it is my right to do it, I cannot be criticised for doing it!" they whine. Witness Limbaugh's own temper trantrum when he discovered that after years of repeated race-baiting he found it impossible to buy an NFL team with an abundance of black players.
Limbaugh's comment is the logical (well, logical based on his own illogical axioms) corollary to the above. If marriage carries with it great responsibility (and on that narrow point at least I agree with him), then it simply cannot be a right. How could it be? If it was a right, then we could marry whomever we wanted at a drop of a hat without anyone objecting or criticising. That's our right. We could be polygamists. That's Our Right. We could marry someone on Tuesday, divorce them Wednesday, and then marry a duck on Thursday. And then shoot the duck and eat it. And then tell everyone the Democrats did it. OUR RIGHTS ARE LEGION!
It's kind of interesting to see someone with no capacity for self-criticism (or even self-awareness in general) come at this from the opposite angle they normally take, but both directions are mired in the same inability to recognise rights and responsibilities are not mutually exclusive, but rather go hand in hand.
 Just to head off potential criticism, I am not suggesting marrying someone on a whim is wrong, certainly not in the sense that inciting religious hatred or lying to the country to justify lining your own pockets is wrong. I would however say there are obvious potential problems such a move might cause, and the point remains that the fact it can be done does not mean there can be no criticism of it once it has been done.