From Matthew Henry:
http://www.studylight.org/com/mhc-com/view.cgi?book=1co&chapter=007
1. To the advice itself, which is that if an unbelieving husband or wife were pleased to dwell with a Christian relative, the other should not separate. The husband should not put away an unbelieving wife, nor the wife leave an unbelieving husband, 1 Corinthians 7:12,13. The Christian calling did not dissolve the marriage covenant, but bind it the faster, by bringing it back to the original institution, limiting it to two persons, and binding them together for life. The believer is not by faith in Christ loosed from matrimonial bonds to an unbeliever, but is at once bound and made apt to be a better relative. But, though a believing wife or husband should not separate from an unbelieving mate, yet if the unbelieving relative desert the believer, and no means can reconcile to a cohabitation, in such a case a brother or sister is not in bondage (1 Corinthians 7:15), not tied up to the unreasonable humour, and bound servilely to follow or cleave to the malicious deserter, or not bound to live unmarried after all proper means for reconciliation have been tried, at least of the deserter contract another marriage or be guilty of adultery, which was a very easy supposition, because a very common instance among the heathen inhabitants of Corinth.
In such a case the deserted person must be free to marry again, and it is granted on all hands. And some think that such a malicious desertion is as much a dissolution of the marriage-covenant as death itself. For how is it possible that the two shall be one flesh when the one is maliciously bent to part from or put away the other? Indeed, the deserter seems still bound by the matrimonial contract; and therefore the apostle says (1 Corinthians 7:11), If the woman depart from her husband upon the account of his infidelity, let her remain unmarried.
But the deserted party seems to be left more at liberty (I mean supposing all the proper means have been used to reclaim the deserter, and other circumstances make it necessary) to marry another person. It does not seem reasonable that they should be still bound, when it is rendered impossible to perform conjugal duties or enjoy conjugal comforts, through the mere fault of their mate: in such a case marriage would be a state of servitude indeed. But, whatever liberty be indulged Christians in such a case as this, they are not allowed, for the mere infidelity of a husband or wife, to separate; but, if the unbeliever be willing, they should continue in the relation, and cohabit as those who are thus related. This is the apostle's general direction.
Adam Clarke:
http://www.studylight.org/com/acc/view.cgi?book=1co&chapter=007
Divorces cannot be allowed
but in the case of fornication: an act of this kind dissolves the marriage vow; but nothing else can. It is a fact that, among the Jews, the wife had just as much right to put away her husband as the husband had to put away his wife. As divorces were granted, it was right that each should have an equal power; for this served as a mutual check.
Whether husband or wife: if such obstinately depart and utterly refuse all cohabitation, a brother or a sister-a Christian man or woman,
is not under bondage to any particular laws, so as to be prevented from remarrying. Such, probably, the law stood then; but it is not so now; for the marriage can only be dissolved by death, or by the ecclesiastical court. Even fornication or adultery does not dissolve the marriage contract; nor will the obstinate separation of any of the parties, however long continued, give the party abandoned authority to remarry. If the person have been beyond sea, and not heard of for seven years, it is presumed he may be dead; and marriage has been connived at in such cases. If there be no person to complain, it may be presumed that there is none injured. But I have known instances where even a marriage after seven years' absence has been very unfortunate; the husband returning at the end of ten or twelve years, and to his utter distress finding his wife married to another man, and with issue of that marriage! There can be no safety in this case, unless there be absolute certainty of the death of the party in question.
