Its indissolubility consists in the fact that the reciprocal bonds contracted in marriage can only be broken by death. God established it in this way from the beginning, when He instituted marriage in the earthly Paradise.
“Therefore, now they are not two, but one flesh. What therefore God hath joined together, let no man put asunder” (Mt. 19:6).
“Whosoever shall put away his wife and marry another, committeth adultery against her. And if the wife shall put away her husband, and be married to another, she committeth adultery” (Mk. 10:11-12).
“Everyone that putteth away his wife, and marrieth another, committeth adultery: and he that marrieth her that is put away from her husband, committeth adultery” (Lk. 16:18).
A marriage between baptized Catholics, once contracted and consummated, is indissoluble. If it was only contracted, it can be dissolved de jure by a solemn religious profession or by a special dispensation from the Holy See.
A marriage between two infidels or non-baptized persons is naturally indissoluble.
The civil authority can in no way break this bond of Christian marriage, for it has no competence over the sacraments and it cannot separate that which God has united. Civil marriage is nothing more than a pure formality prescribed by the law to grant and ensure the civil effects of marriage to the spouses and their children.
Christian spouses who live together although only united by civil marriage would be in a habitual state of mortal sin, and their union would remain illegitimate before God and the Church.