This is completely wrong. Gingrich has a very real problem with his several marriages, despite them being contracted, presumably, not in a Catholic church (Roman or otherwise).
The Catholic Church presumes every apparent marriage to be sacramentally valid. In order to proclaim a marriage invalid, a civil divorce is not enough: a defect should be found in how the putative marriage was contracted. But for a Protestant to marry in a Baptist or any other church, or by a civil ceremony is not a defect. He may have a great difficulty annuling either former marriage, and regularizing the present one.
It is in fact the opposite: if a Catholic (Roman, Melkite, Maronite, or what have you) contracts a marriage outside of the Catholic Church without a dispensation from his bishop, that already invalidates it. Not so when no party is Catholic, then the marriage is harder to invalidate because there is no canonical form for non-Catholics.
My wife works at the local diocesan office and works in the tribunal office that handles annulments and the like. What you wrote sounds a lot of what she has told me from time to time. She’s a Roman Catholic convert herself and loves being one.
Which brings us back to the question: how does he get around the little inconvenience of marriage being a Sacrament?
The marriages do pose a problem, but without knowing the baptismal status of his former spouses; it is possible he could apply for a Pauline or Petrine privilege, and then marry within the church.
When Christians of all stripesshow the same disdain for divorce that Christ and Paul did, then we’ll have true children of God. How about converting and NOT marrying again so as to avoid the judgement of adultery?