Minutiae of Papal infallibility aside, we believe as Catholics that the Holy Spirit protects the Church as a whole from errors in teaching on faith and morals. The Catholic Church is based upon the belief that it is the visible Body of Christ on Earth. It's authority comes from the authority of the Apostles, as appointed by Christ.
So what happens when the Catholic Church reverses itself on matters of faith and morals? It ceases to be the Catholic Church. When Christ instituted the Church in Matthew 16 the sacred author records
Jesus said to him in reply, “Blessed are you, Simon son of Jonah. For flesh and blood has not revealed this to you, but my heavenly Father.
And so I say to you, you are Peter, and upon this rock I will build my church, and the gates of the netherworld shall not prevail against it.
I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven.”That is a weighty responsibility! For the successor of Peter to say essentially "Peter was wrong about morality, President Obama is right" is to essentially take apart the Church. It means the Apostles were wrong, and they don't have moral authority. Therefore the Church does not have moral authority, since it is derived from being the successors to the Apostles.
In fact, we can go further. Since Jesus' own words gave them that authority and said it would never fail, the failure of the Church's authority means the words of Jesus were wrong. Therefore Christ is not Lord, just a man who was misguided lunatic and who should be ignored.
This may sound like a ridiculously overblown conclusion, but if we follow the statement to its logical conclusion, that's where it leads. And so, reductio ad absurdum what these people are calling for is the overthrow of Christ as Lord. I'm sure they don't think that's what they want to happen, but as I said, that's where that train takes you.