Objection 1. It would seem that there can be no marriage between unbelievers. For matrimony is a sacrament of the Church. Now Baptism is the door of the sacraments. Therefore unbelievers, since they are not baptized, cannot marry any more than they can receive other sacraments.
Objection 2. Further, two evils are a greater impediment to good than one. But the unbelief of only one party is an impediment to marriage. Much more, therefore, is the unbelief of both, and consequently there can be no marriage between unbelievers.
Objection 3. Further, just as there is disparity of worship between believer and unbeliever, so can there be between two unbelievers, for instance if one be a heathen and the other a Jew. Now disparity of worship is an impediment to marriage, as stated above (Article 1). Therefore there can be no valid marriage at least between unbelievers of different worship.
Objection 4. Further, in marriage there is real chastity. But according to Augustine (De Adult. Conjug. i, 18) there is no real chastity between an unbeliever and his wife, and these words are quoted in the Decretals (XXVIII, qu. i, can. Sic enim.). Neither therefore is there a true marriage.
Objection 5. Further, true marriage excuses carnal intercourse from sin. But marriage contracted between unbelievers cannot do this, since "the whole life of unbelievers is a sin," as a gloss observes on Romans 14:23, "All that is not of faith is sin." Therefore there is no true marriage between unbelievers.
On the contrary, It is written (1 Corinthians 7:12): "If any brother hath a wife that believeth not, and she consent to dwell with him, let him not put her away." But she is not called his wife except by reason of marriage. Therefore marriage between unbelievers is a true marriage.
Further, the removal of what comes after does not imply the removal of what comes first. Now marriage belongs to an office of nature, which precedes the state of grace, the principle of which is faith. Therefore unbelief does not prevent the existence of marriage between unbelievers.
I answer that, Marriage was instituted chiefly for the good of the offspring, not only as to its begetting--since this can be effected even without marriage--but also as to its advancement to a perfect state, because everything intends naturally to bring its effect to perfection. Now a twofold perfection is to be considered in the offspring. one is the perfection of nature, not only as regards the body but also as regards the soul, by those means which are of the natural law. The other is the perfection of grace: and the former perfection is material and imperfect in relation to the latter. Consequently, since those things which are for the sake of the end are proportionate to the end, the marriage that tends to the first perfection is imperfect and material in comparison with that which tends to the second perfection. And since the first perfection can be common to unbelievers and believers, while the second belongs only to believers, it follows that between unbelievers there is marriage indeed, but not perfected by its ultimate perfection as there is between believers.
Reply to Objection 1. Marriage was instituted not only as a sacrament, but also as an office of nature. And therefore, although marriage is not competent to unbelievers, as a sacrament dependent on the dispensation of the Church's ministers, it is nevertheless competent to them as fulfilling an office of nature. And yet even a marriage of this kind is a sacrament after the manner of a habit, although it is not actually since they do not marry actually in the faith of the Church.
Reply to Objection 2. Disparity of worship is an impediment to marriage, not by reason of unbelief, but on account of the difference of faith. For disparity of worship hinders not only the second perfection of the offspring, but also the first, since the parents endeavor to draw their children in different directions, which is not the case when both are unbelievers.
Reply to Objection 3. As already stated (ad 1) there is marriage between unbelievers, in so far as marriage fulfills an office of nature. Now those things that pertain to the natural law are determinable by positive law: and therefore if any law among unbelievers forbid the contracting of marriage with unbelievers of a different rite, the disparity of worship will be an impediment to their intermarrying. They are not, however, forbidden by Divine law, because before God, however much one may stray from the faith, this makes no difference to one's being removed from grace: nor is it forbidden by any law of the Church who has not to judge of those who are without.
Reply to Objection 4. The chastity and other virtues of unbelievers are said not to be real, because they cannot attain the end of real virtue, which is real happiness. Thus we say it is not a real wine if it has not the effect of wine.
Reply to Objection 5. An unbeliever does not sin in having intercourse with his wife, if he pays her the marriage debt, for the good of the offspring, or for the troth whereby he is bound to her: since this is an act of justice and of temperance which observes the due circumstance in pleasure of touch; even as neither does he sin in performing acts of other civic virtues. Again, the reason why the whole life of unbelievers is said to be a sin is not that they sin in every act, but because they cannot be delivered from the bondage of sin by that which they do.
Source: Summa Theologica (New Advent)