I agree there is no mention of remarriage after you let someone leave; I believe you can come to this conclusion by merely
being honest with the text.
In verse 15 (literal translation):
“But if the unbeliever [wishes to be separated] is separating himself, let him separate himself. The brother or the sister has not been enslaved into such. But God has called us [or you] into peace.”
My commentary on this is that it doesn’t use Jesus’ legal term for divorce in this verse, thus it is a “separation.” There is no mention of a “certificate of divorce” in this whole chapter. Marriage is literally not “slavery”. The “bonds of marriage” are not the same as this Greek term for “enslaved”. Bonds and bondage are not the same words in the Greek. Bonds are legal ties that bind you to a “covenant” for example in marriage. Bondage in English has something to do with the chains of slavery, thus these words mean two different things. Thus Bondage refers to enslavement, not the legal ties or “bonds” of marriage. Luther came up with the excuse of divorce and remarriage for “desertion” unfortunately. This was never an excuse for a “certificate of divorce”. The early church viewed infidelity in marriage to be dealt with as Paul taught to do with the man who “had” his father’s wife in 1 Cor. 5:1- in that they needed to “expel” the impenitent believer from their midst lest the “leaven” of sin would affect them. However, notice in chapter 5:5, the hope was that “turning him over to Satan for the destruction of the flesh” would cause the “sinner” to repent and return to the Lord and thus be reinstated into fellowship. “Putting away” or “separating themselves” was never thought to be permanent, because their hope and prayer was for the full restoration of the wayward believer, or spouse, into fellowship, as should we!
***So, what did Paul say that the Corinthians were supposed to do if their believing spouse was wanting to “separate/depart” (Strong’s 5563) from them: they were to remain unmarried or be reconciled 1 Cor. 7:11. And if an unbelieving spouse wanted to “separate/depart” from a believing spouse: let them “separate/depart” 1 Cor. 7:15 Notice, there is no mention of getting remarried to someone else. Actually, it only says “a brother or sister is literally ‘in no way’ enslaved in such.” The translators often use the term “bondage” instead of “enslaved” which can be confused (purposely?) with the word “bonds” of marriage … which is a different word and meaning. In fact the last phrase of 7:15 says: “But God has called us to peace.” “Peace” in the Greek can have the implication of “reconciliation in relationship” (Strong’s g1515/1518).
In fact it doesn’t use the legal term for “divorce” in this whole chapter, the term that Jesus used in Matthew, Mark, and Luke (Strong’s number g630). Instead the terms “separate/depart” and “leave” (Strong’s g863) are the only terms used in this chapter, not like most translators translate them to be, which is unfortunately usually translated as “divorce.”
Why we know that Paul would not be thinking about spouses being freed from their enslavement to remarry, is what he says at the end of that same chapter in verse 39. It says that the wife wasn’t free to remarry until after the death of her husband. And Paul in Romans 7:2-3 says that the wife becomes an adulteress if she marries while her husband is still living. Jesus stated the permanency of marriage in His words “the two become one.” Jesus said the command that “no man put asunder (literally) the "yoke” (another Greek term for married union)."